From 4203b93717eecf870cf837267a03ed92343e1e0c Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Fri, 18 Apr 2025 17:58:12 +0200
Subject: [PATCH 01/32] Create PseudoRiemannian

This PR is divided in 3 parts. Next is Riemannian Metric (with implementation) and then is Levi-Civita Connection
---
 PhysLean.lean                                 |   1 +
 .../Metric/PseudoRiemannian/Defs.lean         | 425 ++++++++++++++++++
 2 files changed, 426 insertions(+)
 create mode 100644 PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean

diff --git a/PhysLean.lean b/PhysLean.lean
index 40d56af17..2b111421d 100644
--- a/PhysLean.lean
+++ b/PhysLean.lean
@@ -15,6 +15,7 @@ import PhysLean.Mathematics.LinearMaps
 import PhysLean.Mathematics.List
 import PhysLean.Mathematics.List.InsertIdx
 import PhysLean.Mathematics.List.InsertionSort
+import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
 import PhysLean.Mathematics.PiTensorProduct
 import PhysLean.Mathematics.RatComplexNum
 import PhysLean.Mathematics.SO3.Basic
diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
new file mode 100644
index 000000000..d4c5fead8
--- /dev/null
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -0,0 +1,425 @@
+import Mathlib.Analysis.InnerProductSpace.Basic
+import Mathlib.Geometry.Manifold.VectorBundle.SmoothSection
+import Mathlib.LinearAlgebra.BilinearForm.Properties
+import Mathlib.LinearAlgebra.QuadraticForm.Basic
+
+/-!
+# Pseudo-Riemannian Metrics on Smooth Manifolds
+
+This file formalizes the fundamental notion of pseudo-Riemannian metrics on smooth manifolds
+and establishes their basic properties. A pseudo-Riemannian metric equips a manifold with a
+smoothly varying, non-degenerate, symmetric bilinear form on each tangent space, generalizing
+the concept of an inner product space to curved spaces.
+
+## Main Definitions
+
+* `PseudoRiemannianMetric I n M`: A structure representing a `C^n` pseudo-Riemannian metric
+  on a manifold `M` modelled on `(E, H)` with model with corners `I`. It consists of a family
+  of non-degenerate, symmetric, continuous bilinear forms `gₓ` on each tangent space `TₓM`,
+  varying `C^n`-smoothly with `x`. The base field `𝕜` is required to be `RCLike`, the model
+  space `E` must be finite-dimensional, and the manifold `M` must be `C^{n+1}` smooth.
+
+* `PseudoRiemannianMetric.flatEquiv g x`: The "musical isomorphism" from the tangent space at `x`
+  to its dual space, representing the canonical isomorphism induced by the metric.
+
+* `PseudoRiemannianMetric.sharpEquiv g x`: The inverse of the flat isomorphism, mapping from
+  the dual space back to the tangent space.
+
+## Implementation Notes
+
+* The bilinear form `gₓ` at a point `x` is represented as a `ContinuousLinearMap`
+  `val x : TₓM →L[𝕜] (TₓM →L[𝕜] 𝕜)`. This curried form `v ↦ (w ↦ gₓ(v, w))` encodes both
+  the bilinear structure and its continuity.
+
+* Smoothness is defined by requiring that for any two `C^n` vector fields `X, Y`, the scalar
+  function `x ↦ gₓ(X x, Y x)` is `C^n` smooth (`ContMDiff n`).
+
+* The manifold `M` must be `C^{n+1}` smooth (`IsManifold I (n + 1) M`) to ensure the tangent
+  bundle is a `C^n` vector bundle, making the notion of `C^n` vector fields well-defined.
+
+* Finite-dimensionality of the model space (`FiniteDimensional 𝕜 E`) guarantees that tangent
+  spaces are finite-dimensional, which is necessary for establishing that non-degeneracy implies
+  the existence of an isomorphism between the tangent space and its dual.
+-/
+
+section PseudoRiemannianMetric
+
+noncomputable section
+
+universe u v w
+
+open Bundle Set Function Filter Module Topology ContinuousLinearMap
+open scoped Manifold Bundle LinearMap Dual
+
+/-- Convert a continuous linear map representing a bilinear form to a `LinearMap.BilinForm`. -/
+def ContinuousLinearMap.toBilinForm
+    {𝕜 E : Type*} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E]
+    (f : E →L[𝕜] (E →L[𝕜] 𝕜)) : LinearMap.BilinForm 𝕜 E :=
+  LinearMap.mk₂ 𝕜 (fun v w => (f v) w)
+    (fun v₁ v₂ w => by dsimp; rw [f.map_add]; rfl)
+    (fun a v w => by dsimp; rw [f.map_smul]; rfl)
+    (fun v w₁ w₂ => by dsimp; rw [(f v).map_add];)
+    (fun a v w => (f v).map_smul a w)
+
+-- We use Real-like fields for InnerProductSpace in Riemannian Metric
+variable {𝕜 : Type u} [RCLike 𝕜] -- Stronger assumption for InnerProductSpace, implies NontriviallyNormedField
+variable {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] -- Requires finite dimension on the model space
+variable {H : Type w} [TopologicalSpace H] -- Chart space
+variable {M : Type w} [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E] -- Manifold M and ChartedSpace for E
+variable {I : ModelWithCorners 𝕜 E H} {n : WithTop ℕ∞} -- Model with corners and smoothness level
+
+/-- A pseudo-Riemannian metric of smoothness class `C^n` on a manifold `M` modelled on `(E, H)`
+    with model `I`. This structure defines a smoothly varying, non-degenerate, symmetric,
+    continuous bilinear form `gₓ` on the tangent space `TₓM` at each point `x`.
+    Requires `M` to be `C^{n+1}` smooth.
+-/
+@[ext]
+structure PseudoRiemannianMetric
+    (I : ModelWithCorners 𝕜 E H) (n : WithTop ℕ∞)
+    -- Instances ensuring the tangent bundle is a well-defined smooth vector bundle
+    [TopologicalSpace (TangentBundle I M)]
+    [FiberBundle E (TangentSpace I : M → Type _)]
+    [VectorBundle 𝕜 E (TangentSpace I : M → Type _)]
+    [IsManifold I (n + 1) M] -- Manifold is C^{n+1}
+    [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I] -- Tangent bundle is C^n
+   : Type (max u v w) where
+  /-- The metric tensor at each point `x : M`, represented as a continuous linear map
+      `TₓM →L[𝕜] (TₓM →L[𝕜] 𝕜)`. Applying it twice, `(val x v) w`, yields `gₓ(v, w)`. -/
+  protected val : ∀ (x : M), TangentSpace I x →L[𝕜] (TangentSpace I x →L[𝕜] 𝕜)
+  /-- The metric is symmetric: `gₓ(v, w) = gₓ(w, v)`. -/
+  protected symm : ∀ (x : M) (v w : TangentSpace I x), (val x v) w = (val x w) v
+  /-- The metric is non-degenerate: if `gₓ(v, w) = 0` for all `w`, then `v = 0`. -/
+  protected nondegenerate : ∀ (x : M) (v : TangentSpace I x), (∀ w : TangentSpace I x, (val x v) w = 0) → v = 0
+  /-- The metric varies smoothly: `x ↦ gₓ(Xₓ, Yₓ)` is a `C^n` function for `C^n` vector fields `X, Y`. -/
+  protected smooth : ∀ (X Y : ContMDiffSection I E n (TangentSpace I)),
+    ContMDiff I (modelWithCornersSelf 𝕜 𝕜) n (fun x => val x (X x) (Y x))
+
+namespace PseudoRiemannianMetric
+
+instance TangentSpace.addCommGroup (x : M) : AddCommGroup (TangentSpace I x) := by infer_instance
+instance TangentSpace.module (x : M) : Module 𝕜 (TangentSpace I x) := by infer_instance
+
+/-- If two vector spaces are linearly equivalent, and one is finite-dimensional,
+    then so is the other. This version transfers the finite-dimensional property
+    via a continuous linear equivalence. -/
+lemma FiniteDimensional.of_equiv
+    {𝕜 : Type u} [RCLike 𝕜]
+    {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E]
+    {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F]
+    [FiniteDimensional 𝕜 E]
+    (e : E ≃L[𝕜] F) : FiniteDimensional 𝕜 F :=
+  LinearEquiv.finiteDimensional e.toLinearEquiv
+
+/-- If two vector spaces are linearly equivalent, and one is finite-dimensional,
+    then so is the other. This version works with a linear equivalence. -/
+lemma FiniteDimensional.of_linearEquiv
+    {𝕜 : Type u} [RCLike 𝕜]
+    {E : Type v} [AddCommGroup E] [Module 𝕜 E]
+    {F : Type w} [AddCommGroup F] [Module 𝕜 F]
+    [FiniteDimensional 𝕜 E]
+    (e : E ≃ₗ[𝕜] F) : FiniteDimensional 𝕜 F := by
+  exact Finite.equiv e
+
+lemma VectorBundle.finiteDimensional_fiber
+    (𝕜 : Type u) [RCLike 𝕜]
+    {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F]
+    {B : Type w} [TopologicalSpace B]
+    {E_bundle : B → Type*} [∀ x, AddCommGroup (E_bundle x)] [∀ x, Module 𝕜 (E_bundle x)]
+    [TopologicalSpace (Bundle.TotalSpace F E_bundle)]
+    [∀ x, TopologicalSpace (E_bundle x)]
+    [FiberBundle F E_bundle] [VectorBundle 𝕜 F E_bundle]
+    (x : B) : FiniteDimensional 𝕜 (E_bundle x) :=
+  let triv := trivializationAt F E_bundle x
+  let hx := mem_baseSet_trivializationAt F E_bundle x
+  have h_linear : triv.IsLinear 𝕜 := VectorBundle.trivialization_linear' triv
+  haveI : triv.IsLinear 𝕜 := h_linear
+  let linear_equiv := triv.continuousLinearEquivAt 𝕜 x hx
+  FiniteDimensional.of_equiv linear_equiv.symm
+
+instance TangentSpace.finiteDimensional
+  [FiniteDimensional 𝕜 E]
+  [∀ x : M, TopologicalSpace (TangentSpace I x)]
+  [TopologicalSpace (Bundle.TotalSpace E (TangentSpace I : M → Type _))]
+  [FiberBundle E (TangentSpace I : M → Type _)]
+  [VectorBundle 𝕜 E (TangentSpace I : M → Type _)]
+  (x : M) :
+    FiniteDimensional 𝕜 (TangentSpace I x) :=
+  VectorBundle.finiteDimensional_fiber 𝕜 (F := E) (E_bundle := TangentSpace I) x
+
+variable
+    [inst_top : TopologicalSpace (TangentBundle I M)]
+    [inst_fib : FiberBundle E (TangentSpace I : M → Type _)]
+    [inst_vec : VectorBundle 𝕜 E (TangentSpace I : M → Type _)]
+    [inst_mani : IsManifold I (n + 1) M]
+    [inst_cmvb : ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
+
+/-- Convert the continuous linear map representation `val x` to the algebraic `LinearMap.BilinForm`.
+    This is useful for leveraging lemmas about `BilinForm`. -/
+def toBilinForm
+    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
+         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
+    LinearMap.BilinForm 𝕜 (TangentSpace I x) :=
+  (g.val x).toBilinForm
+
+/-- Coercion from PseudoRiemannianMetric to its function representation. -/
+instance : CoeFun (@PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
+        inst_fib inst_vec inst_mani inst_cmvb)
+        (fun _ => ∀ x : M, TangentSpace I x →L[𝕜] (TangentSpace I x →L[𝕜] 𝕜)) where
+           coe g := g.val
+
+omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
+@[simp] lemma toBilinForm_apply
+    (g : PseudoRiemannianMetric I n) (x : M) (v w : TangentSpace I x) :
+  g.toBilinForm x v w = g.val x v w := rfl
+
+omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
+@[simp] lemma toBilinForm_isSymm
+    (g : PseudoRiemannianMetric I n) (x : M) : (g.toBilinForm x).IsSymm := by
+  intro v w
+  simp only [toBilinForm_apply]
+  exact g.symm x v w
+
+omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
+@[simp] lemma toBilinForm_nondegenerate
+    (g : PseudoRiemannianMetric I n) (x : M) :
+    (g.toBilinForm x).Nondegenerate := by
+  intro v hv
+  simp_rw [toBilinForm_apply] at hv
+  exact g.nondegenerate x v hv
+
+omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
+lemma symm'
+    (g : PseudoRiemannianMetric I n) (x : M) (v w : TangentSpace I x) :
+    (g.val x v) w = (g.val x w) v :=
+  g.symm x v w
+
+omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
+lemma nondegenerate'
+    (g : PseudoRiemannianMetric I n) (x : M) (v : TangentSpace I x)
+    (h : ∀ w : TangentSpace I x, (g.val x v) w = 0) : v = 0 :=
+  g.nondegenerate x v h
+
+omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
+lemma smooth'
+    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
+         inst_fib inst_vec inst_mani inst_cmvb) (X Y : ContMDiffSection I E n (TangentSpace I)) :
+    ContMDiff I (modelWithCornersSelf 𝕜 𝕜) n (fun x => (g.val x (X x)) (Y x)) :=
+  g.smooth X Y
+
+/-- Convert a pseudo-Riemannian metric at a point to a quadratic form. -/
+@[simp] def toQuadraticForm
+    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
+         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
+    QuadraticForm 𝕜 (TangentSpace I x) where
+  toFun v := g.val x v v
+  toFun_smul a v := by
+    simp only [ContinuousLinearMap.map_smul, ContinuousLinearMap.smul_apply, smul_smul, pow_two]
+  exists_companion' :=
+      ⟨ LinearMap.mk₂ 𝕜 (fun v y => g.val x v y + g.val x y v)
+        (fun v₁ v₂ y => by -- Additivity in v
+          simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.add_apply]
+          ring)
+        (fun a v y => by -- Homogeneity in v
+          simp only [ContinuousLinearMap.map_smul, ContinuousLinearMap.smul_apply]
+          ring_nf
+          exact Eq.symm (DistribSMul.smul_add a (((g.val x) v) y) (((g.val x) y) v)))
+        (fun v y₁ y₂ => by -- Additivity in y
+          simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.add_apply]
+          ring)
+        (fun a v y => by -- Homogeneity in y
+          simp only [ContinuousLinearMap.map_smul, ContinuousLinearMap.smul_apply]
+          ring_nf
+          exact Eq.symm (DistribSMul.smul_add a (((g.val x) v) y) (((g.val x) y) v))),
+            by
+              intro v y
+              simp only [LinearMap.mk₂_apply, ContinuousLinearMap.map_add,
+                ContinuousLinearMap.add_apply, g.symm x]
+              ring⟩
+
+omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
+lemma toQuadraticForm_apply
+    (g : PseudoRiemannianMetric I n) (x : M) (v : TangentSpace I x) :
+    g.toQuadraticForm x v = g.val x v v := rfl
+
+/-- The "musical" isomorphism (index lowering) from the tangent space to its dual,
+    induced by a pseudo-Riemannian metric. -/
+def flat
+    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
+         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
+    TangentSpace I x →ₗ[𝕜] (TangentSpace I x →L[𝕜] 𝕜) :=
+  { toFun := λ v => g.val x v,
+    map_add' := λ v w => by simp only [ContinuousLinearMap.map_add],
+    map_smul' := λ a v => by simp only [ContinuousLinearMap.map_smul]; exact rfl }
+
+omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
+@[simp] lemma flat_apply
+    (g : PseudoRiemannianMetric I n) (x : M) (v w : TangentSpace I x) :
+    (g.flat x v) w = g.val x v w := rfl
+
+/-- The musical isomorphism as a continuous linear map. -/
+def flatL
+    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
+         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
+    TangentSpace I x →L[𝕜] (TangentSpace I x →L[𝕜] 𝕜) :=
+  { g.flat x with
+    cont := ContinuousLinearMap.continuous (g.val x) }
+
+omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
+@[simp] lemma flatL_apply
+    (g : PseudoRiemannianMetric I n) (x : M) (v w : TangentSpace I x) :
+    (g.flatL x v) w = g.val x v w := rfl
+
+omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
+/-- The linear map `flat` is injective due to non-degeneracy. -/
+@[simp] lemma flat_inj
+    (g : PseudoRiemannianMetric I n) (x : M) :
+    Function.Injective (g.flat x) := by
+  rw [← LinearMap.ker_eq_bot]
+  apply LinearMap.ker_eq_bot'.mpr
+  intro v hv
+  apply g.nondegenerate' x v
+  intro w
+  exact DFunLike.congr_fun hv w
+
+omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
+/-- The continuous linear map `flatL` is injective. -/
+@[simp] lemma flatL_inj
+    (g : PseudoRiemannianMetric I n) (x : M) :
+    Function.Injective (g.flatL x) := by
+  intro v₁ v₂ h
+  apply flat_inj g x
+  exact h
+
+omit [ChartedSpace H E] in
+/-- The continuous linear map `flatL` is surjective because the tangent space is finite dimensional. -/
+@[simp] lemma flatL_surj
+    (g : PseudoRiemannianMetric I n) (x : M) :
+    Function.Surjective (g.flatL x) := by
+  haveI : FiniteDimensional 𝕜 (TangentSpace I x) := TangentSpace.finiteDimensional x
+  have h_finrank_eq : finrank 𝕜 (TangentSpace I x) = finrank 𝕜 (TangentSpace I x →L[𝕜] 𝕜) := by
+    have h_dual_eq : finrank 𝕜 (TangentSpace I x →L[𝕜] 𝕜) = finrank 𝕜 (Module.Dual 𝕜 (TangentSpace I x)) := by
+      let to_dual : (TangentSpace I x →L[𝕜] 𝕜) → Module.Dual 𝕜 (TangentSpace I x) := fun f => f.toLinearMap
+      let from_dual : Module.Dual 𝕜 (TangentSpace I x) → (TangentSpace I x →L[𝕜] 𝕜) := fun f =>
+        ContinuousLinearMap.mk f (by
+          apply LinearMap.continuous_of_finiteDimensional)
+      let equiv : (TangentSpace I x →L[𝕜] 𝕜) ≃ₗ[𝕜] Module.Dual 𝕜 (TangentSpace I x) :=
+      { toFun := to_dual,
+        invFun := from_dual,
+        map_add' := fun f g => by
+          ext v; unfold to_dual; simp only [LinearMap.add_apply]; rfl,
+        map_smul' := fun c f => by
+          ext v; unfold to_dual; simp only [LinearMap.smul_apply]; rfl,
+        left_inv := fun f => by
+          ext v; unfold to_dual from_dual; simp,
+        right_inv := fun f => by
+          ext v; unfold to_dual from_dual; simp }
+      exact LinearEquiv.finrank_eq equiv
+    rw [h_dual_eq, ← Subspace.dual_finrank_eq]
+  exact (LinearMap.injective_iff_surjective_of_finrank_eq_finrank h_finrank_eq).mp (flatL_inj g x)
+
+/-- The "musical" isomorphism (index lowering) from the tangent space to its dual,
+    as a continuous linear equivalence. -/
+@[simp] def flatEquiv
+    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
+         inst_fib inst_vec inst_mani inst_cmvb)
+    (x : M) :
+    TangentSpace I x ≃L[𝕜] (TangentSpace I x →L[𝕜] 𝕜) :=
+  LinearEquiv.toContinuousLinearEquiv
+    (LinearEquiv.ofBijective
+      ((g.flatL x).toLinearMap)
+      ⟨g.flatL_inj x, g.flatL_surj x⟩)
+
+omit [ChartedSpace H E] in
+lemma coe_flatEquiv
+    (g : PseudoRiemannianMetric I n) (x : M) :
+    (g.flatEquiv x : TangentSpace I x →ₗ[𝕜] (TangentSpace I x →L[𝕜] 𝕜)) = g.flatL x := rfl
+
+omit [ChartedSpace H E] in
+@[simp] lemma flatEquiv_apply
+    (g : PseudoRiemannianMetric I n) (x : M) (v w : TangentSpace I x) :
+    (g.flatEquiv x v) w = g.val x v w := rfl
+
+/-- The "musical" isomorphism (index raising) from the dual of the tangent space to the tangent space,
+    induced by a pseudo-Riemannian metric. This is the inverse of `flatEquiv`. -/
+def sharpEquiv
+    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
+         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
+    (TangentSpace I x →L[𝕜] 𝕜) ≃L[𝕜] TangentSpace I x :=
+  (g.flatEquiv x).symm
+#lint
+/-- The index raising map `sharp` as a continuous linear map. -/
+def sharpL
+    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
+         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
+    (TangentSpace I x →L[𝕜] 𝕜) →L[𝕜] TangentSpace I x :=
+  (g.sharpEquiv x).toContinuousLinearMap
+
+omit [ChartedSpace H E] in
+lemma sharpL_eq_toContinuousLinearMap
+  (g : PseudoRiemannianMetric I n) (x : M) :
+  g.sharpL x = (g.sharpEquiv x).toContinuousLinearMap := rfl
+
+omit [ChartedSpace H E] in
+lemma coe_sharpEquiv
+    (g : PseudoRiemannianMetric I n) (x : M) :
+    (g.sharpEquiv x : (TangentSpace I x →L[𝕜] 𝕜) →L[𝕜] TangentSpace I x) = g.sharpL x := rfl
+
+/-- The index raising map `sharp` as a linear map. -/
+noncomputable def sharp
+    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
+         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
+    (TangentSpace I x →L[𝕜] 𝕜) →ₗ[𝕜] TangentSpace I x :=
+  (g.sharpEquiv x).toLinearEquiv.toLinearMap
+
+omit [ChartedSpace H E] in
+@[simp] lemma sharpL_apply_flatL
+    (g : PseudoRiemannianMetric I n) (x : M) (v : TangentSpace I x) :
+    g.sharpL x (g.flatL x v) = v :=
+  (g.flatEquiv x).left_inv v
+
+omit [ChartedSpace H E] in
+@[simp] lemma flatL_apply_sharpL
+    (g : PseudoRiemannianMetric I n) (x : M) (ω : TangentSpace I x →L[𝕜] 𝕜) :
+    g.flatL x (g.sharpL x ω) = ω :=
+  (g.flatEquiv x).right_inv ω
+
+omit [ChartedSpace H E] in
+/-- Applying `sharp` then `flat` recovers the original covector. -/
+@[simp] lemma flat_sharp_apply
+    (g : PseudoRiemannianMetric I n) (x : M) (ω : TangentSpace I x →L[𝕜] 𝕜) :
+    g.flat x (g.sharp x ω) = ω := by
+  -- Use the continuous versions and coerce
+  have := flatL_apply_sharpL g x ω
+  -- Need to relate flat and flatL, sharp and sharpL
+  simp only [sharp, sharpL, flat, flatL, coe_flatEquiv]; simp only [coe_sharpEquiv,
+             ContinuousLinearEquiv.coe_coe, LinearEquiv.coe_coe] at this ⊢
+  exact this
+
+omit [ChartedSpace H E] in
+@[simp] lemma sharp_flat_apply
+    (g : PseudoRiemannianMetric I n) (x : M) (v : TangentSpace I x) :
+    g.sharp x (g.flat x v) = v := by
+  -- Use the continuous versions and coerce
+  have := sharpL_apply_flatL g x v
+  simp only [sharp, sharpL, flat, flatL]; simp only [coe_flatEquiv, coe_sharpEquiv,
+             ContinuousLinearEquiv.coe_coe, LinearEquiv.coe_coe] at this ⊢
+  exact this
+
+omit [ChartedSpace H E] in
+/-- The metric evaluated at `sharp ω₁` and `sharp ω₂`. -/
+@[simp] lemma apply_sharp_sharp
+    (g : PseudoRiemannianMetric I n) (x : M) (ω₁ ω₂ : TangentSpace I x →L[𝕜] 𝕜) :
+    g.val x (g.sharpL x ω₁) (g.sharpL x ω₂) = ω₁ (g.sharpL x ω₂) := by
+  rw [← flatL_apply g x (g.sharpL x ω₁)]
+  rw [flatL_apply_sharpL g x ω₁]
+
+omit [ChartedSpace H E] in
+/-- The metric evaluated at `v` and `sharp ω`. -/
+lemma apply_vec_sharp
+    (g : PseudoRiemannianMetric I n) (x : M) (v : TangentSpace I x) (ω : TangentSpace I x →L[𝕜] 𝕜) :
+    g.val x v (g.sharpL x ω) = ω v := by
+  rw [g.symm' x v (g.sharpL x ω)]
+  rw [← flatL_apply g x (g.sharpL x ω)]
+  rw [flatL_apply_sharpL g x ω]
+
+end PseudoRiemannianMetric

From 2341670f400e4dd3c5cf280f3cd6091b03772d43 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Fri, 18 Apr 2025 18:10:17 +0200
Subject: [PATCH 02/32] Update Defs.lean

---
 .../Metric/PseudoRiemannian/Defs.lean         | 40 ++++++++++++-------
 1 file changed, 26 insertions(+), 14 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index d4c5fead8..9be99a29a 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -1,3 +1,9 @@
+/-
+Copyright (c) 2025 Matteo Cipollina. All rights reserved. -- Adjusted year
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Matteo Cipollina
+-/
+
 import Mathlib.Analysis.InnerProductSpace.Basic
 import Mathlib.Geometry.Manifold.VectorBundle.SmoothSection
 import Mathlib.LinearAlgebra.BilinearForm.Properties
@@ -62,10 +68,11 @@ def ContinuousLinearMap.toBilinForm
     (fun a v w => (f v).map_smul a w)
 
 -- We use Real-like fields for InnerProductSpace in Riemannian Metric
-variable {𝕜 : Type u} [RCLike 𝕜] -- Stronger assumption for InnerProductSpace, implies NontriviallyNormedField
-variable {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] -- Requires finite dimension on the model space
+variable {𝕜 : Type u} [RCLike 𝕜] -- Stronger assumption, implies NontriviallyNormedField
+variable {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E]
 variable {H : Type w} [TopologicalSpace H] -- Chart space
-variable {M : Type w} [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E] -- Manifold M and ChartedSpace for E
+-- Manifold M and ChartedSpace for E
+variable {M : Type w} [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E]
 variable {I : ModelWithCorners 𝕜 E H} {n : WithTop ℕ∞} -- Model with corners and smoothness level
 
 /-- A pseudo-Riemannian metric of smoothness class `C^n` on a manifold `M` modelled on `(E, H)`
@@ -81,16 +88,18 @@ structure PseudoRiemannianMetric
     [FiberBundle E (TangentSpace I : M → Type _)]
     [VectorBundle 𝕜 E (TangentSpace I : M → Type _)]
     [IsManifold I (n + 1) M] -- Manifold is C^{n+1}
-    [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I] -- Tangent bundle is C^n
-   : Type (max u v w) where
+    [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I] : -- Tangent bundle is C^n
+    Type (max u v w) where
   /-- The metric tensor at each point `x : M`, represented as a continuous linear map
       `TₓM →L[𝕜] (TₓM →L[𝕜] 𝕜)`. Applying it twice, `(val x v) w`, yields `gₓ(v, w)`. -/
   protected val : ∀ (x : M), TangentSpace I x →L[𝕜] (TangentSpace I x →L[𝕜] 𝕜)
   /-- The metric is symmetric: `gₓ(v, w) = gₓ(w, v)`. -/
   protected symm : ∀ (x : M) (v w : TangentSpace I x), (val x v) w = (val x w) v
   /-- The metric is non-degenerate: if `gₓ(v, w) = 0` for all `w`, then `v = 0`. -/
-  protected nondegenerate : ∀ (x : M) (v : TangentSpace I x), (∀ w : TangentSpace I x, (val x v) w = 0) → v = 0
-  /-- The metric varies smoothly: `x ↦ gₓ(Xₓ, Yₓ)` is a `C^n` function for `C^n` vector fields `X, Y`. -/
+  protected nondegenerate : ∀ (x : M) (v : TangentSpace I x), (∀ w : TangentSpace I x,
+    (val x v) w = 0) → v = 0
+  /-- The metric varies smoothly: `x ↦ gₓ(Xₓ, Yₓ)` is a `C^n` function for `C^n`
+      vector fields `X, Y`. -/
   protected smooth : ∀ (X Y : ContMDiffSection I E n (TangentSpace I)),
     ContMDiff I (modelWithCornersSelf 𝕜 𝕜) n (fun x => val x (X x) (Y x))
 
@@ -291,14 +300,17 @@ omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
   exact h
 
 omit [ChartedSpace H E] in
-/-- The continuous linear map `flatL` is surjective because the tangent space is finite dimensional. -/
+/-- The continuous linear map `flatL` is surjective because the tangent space is
+    finite dimensional. -/
 @[simp] lemma flatL_surj
     (g : PseudoRiemannianMetric I n) (x : M) :
     Function.Surjective (g.flatL x) := by
   haveI : FiniteDimensional 𝕜 (TangentSpace I x) := TangentSpace.finiteDimensional x
   have h_finrank_eq : finrank 𝕜 (TangentSpace I x) = finrank 𝕜 (TangentSpace I x →L[𝕜] 𝕜) := by
-    have h_dual_eq : finrank 𝕜 (TangentSpace I x →L[𝕜] 𝕜) = finrank 𝕜 (Module.Dual 𝕜 (TangentSpace I x)) := by
-      let to_dual : (TangentSpace I x →L[𝕜] 𝕜) → Module.Dual 𝕜 (TangentSpace I x) := fun f => f.toLinearMap
+    have h_dual_eq : finrank 𝕜 (TangentSpace I x →L[𝕜] 𝕜) = finrank 𝕜 (Module.Dual 𝕜
+    (TangentSpace I x)) := by
+      let to_dual : (TangentSpace I x →L[𝕜] 𝕜) → Module.Dual 𝕜 (TangentSpace I x) :=
+        fun f => f.toLinearMap
       let from_dual : Module.Dual 𝕜 (TangentSpace I x) → (TangentSpace I x →L[𝕜] 𝕜) := fun f =>
         ContinuousLinearMap.mk f (by
           apply LinearMap.continuous_of_finiteDimensional)
@@ -339,8 +351,8 @@ omit [ChartedSpace H E] in
     (g : PseudoRiemannianMetric I n) (x : M) (v w : TangentSpace I x) :
     (g.flatEquiv x v) w = g.val x v w := rfl
 
-/-- The "musical" isomorphism (index raising) from the dual of the tangent space to the tangent space,
-    induced by a pseudo-Riemannian metric. This is the inverse of `flatEquiv`. -/
+/-- The "musical" isomorphism (index raising) from the dual of the tangent space to the
+    tangent space, induced by a pseudo-Riemannian metric. This is the inverse of `flatEquiv`. -/
 def sharpEquiv
     (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
          inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
@@ -356,7 +368,7 @@ def sharpL
 
 omit [ChartedSpace H E] in
 lemma sharpL_eq_toContinuousLinearMap
-  (g : PseudoRiemannianMetric I n) (x : M) :
+    (g : PseudoRiemannianMetric I n) (x : M) :
   g.sharpL x = (g.sharpEquiv x).toContinuousLinearMap := rfl
 
 omit [ChartedSpace H E] in
@@ -367,7 +379,7 @@ lemma coe_sharpEquiv
 /-- The index raising map `sharp` as a linear map. -/
 noncomputable def sharp
     (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
-         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
+    inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
     (TangentSpace I x →L[𝕜] 𝕜) →ₗ[𝕜] TangentSpace I x :=
   (g.sharpEquiv x).toLinearEquiv.toLinearMap
 

From 8f44f6cbd24e5532fd971dabeef21710baecd375 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Fri, 18 Apr 2025 18:10:58 +0200
Subject: [PATCH 03/32] Update Defs.lean

---
 .../Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean    | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index 9be99a29a..1da391e6d 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -88,8 +88,8 @@ structure PseudoRiemannianMetric
     [FiberBundle E (TangentSpace I : M → Type _)]
     [VectorBundle 𝕜 E (TangentSpace I : M → Type _)]
     [IsManifold I (n + 1) M] -- Manifold is C^{n+1}
-    [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I] : -- Tangent bundle is C^n
-    Type (max u v w) where
+    -- Tangent bundle is C^n
+    [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I] : Type (max u v w) where
   /-- The metric tensor at each point `x : M`, represented as a continuous linear map
       `TₓM →L[𝕜] (TₓM →L[𝕜] 𝕜)`. Applying it twice, `(val x v) w`, yields `gₓ(v, w)`. -/
   protected val : ∀ (x : M), TangentSpace I x →L[𝕜] (TangentSpace I x →L[𝕜] 𝕜)

From 3f54c98f3c566434172a2f31a653ca4edf5e0f27 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Fri, 18 Apr 2025 18:39:33 +0200
Subject: [PATCH 04/32] Update PhysLean.lean

---
 PhysLean.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/PhysLean.lean b/PhysLean.lean
index 2b111421d..ce57a3c4c 100644
--- a/PhysLean.lean
+++ b/PhysLean.lean
@@ -11,11 +11,11 @@ import PhysLean.Electromagnetism.LorentzAction
 import PhysLean.Electromagnetism.MaxwellEquations
 import PhysLean.Mathematics.Fin
 import PhysLean.Mathematics.Fin.Involutions
+import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
 import PhysLean.Mathematics.LinearMaps
 import PhysLean.Mathematics.List
 import PhysLean.Mathematics.List.InsertIdx
 import PhysLean.Mathematics.List.InsertionSort
-import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
 import PhysLean.Mathematics.PiTensorProduct
 import PhysLean.Mathematics.RatComplexNum
 import PhysLean.Mathematics.SO3.Basic

From d9912f0f42b8fd84240d948d8e949f2ecb2bc5ad Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Fri, 18 Apr 2025 18:47:21 +0200
Subject: [PATCH 05/32] Update Completeness.lean

---
 .../OneDimension/HarmonicOscillator/Completeness.lean | 11 ++++++++---
 1 file changed, 8 insertions(+), 3 deletions(-)

diff --git a/PhysLean/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean b/PhysLean/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean
index 236438539..f048524c6 100644
--- a/PhysLean/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean
+++ b/PhysLean/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean
@@ -124,9 +124,14 @@ lemma mul_power_integrable (f : ℝ → ℂ) (hf : MemHS f) (r : ℕ) :
     suffices h2 : IsUnit (↑((1/Q.ξ)^ r : ℂ)) by
       rw [IsUnit.integrable_smul_iff h2] at h1
       simpa using h1
-    simp only [isUnit_iff_ne_zero, ne_eq, pow_eq_zero_iff', Complex.ofReal_eq_zero, not_and,
-      Decidable.not_not]
-    simp
+    simp only [isUnit_iff_ne_zero, ne_eq, pow_eq_zero_iff', Complex.ofReal_eq_zero]
+    intro h
+    cases h with
+    | intro h =>
+      have h' : Q.ξ = 0 := by
+        simpa [one_div] using h
+      exact Q.ξ_pos.ne' h'
+
   · simp only [ne_eq, Decidable.not_not] at hr
     subst hr
     simpa using Q.mul_physHermite_integrable f hf 0

From 74e85ec85bdb5f72c43e9d720dae985e58fa5495 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Fri, 18 Apr 2025 18:48:18 +0200
Subject: [PATCH 06/32] Update Defs.lean

---
 PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index 1da391e6d..c82f74683 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -1,5 +1,5 @@
 /-
-Copyright (c) 2025 Matteo Cipollina. All rights reserved. -- Adjusted year
+Copyright (c) 2025 Matteo Cipollina. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Matteo Cipollina
 -/

From c9d5940c5732d78e7c46f17ef5e499bbd29d5c54 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Thu, 24 Apr 2025 10:46:22 +0200
Subject: [PATCH 07/32] Create Riemannian

TODO: add InnerProductSpace to complete the section
---
 PhysLean.lean                                 |   1 +
 .../Geometry/Metric/Riemannian/Defs.lean      | 235 ++++++++++++++++++
 .../Metric/PseudoRiemannian/Defs.lean         | 171 +++++++++++++
 3 files changed, 407 insertions(+)
 create mode 100644 PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
 create mode 100644 PhysLean/Relativity/Metric/PseudoRiemannian/Defs.lean

diff --git a/PhysLean.lean b/PhysLean.lean
index ce57a3c4c..e9d0b975b 100644
--- a/PhysLean.lean
+++ b/PhysLean.lean
@@ -12,6 +12,7 @@ import PhysLean.Electromagnetism.MaxwellEquations
 import PhysLean.Mathematics.Fin
 import PhysLean.Mathematics.Fin.Involutions
 import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
+import PhysLean.Mathematics.Geometry.Metric.Riemannian.Defs
 import PhysLean.Mathematics.LinearMaps
 import PhysLean.Mathematics.List
 import PhysLean.Mathematics.List.InsertIdx
diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
new file mode 100644
index 000000000..21b8b5e86
--- /dev/null
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -0,0 +1,235 @@
+/-
+Copyright (c) 2025 Matteo Cipollina. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Matteo Cipollina
+-/
+
+import Mathlib.LinearAlgebra.QuadraticForm.Dual
+import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
+
+/-!
+# Riemannian Metrics on Manifolds
+
+/-!
+This file introduces `RiemannianMetric`, a smooth positive‐definite specialization of
+`PseudoRiemannianMetric` over ℝ.  It requires that each bilinear form `gₓ` satisfy
+`gₓ(v,v) > 0` for nonzero tangent vectors `v`, and uses this to define the canonical
+inner product, norm, and distance on each tangent space.  (TODO: assemble these
+data into an `InnerProductSpace ℝ (TangentSpace I_ℝ x)` instance).
+-/
+
+## Main Definitions
+
+*   `RiemannianMetric I n M`: A structure representing a `C^n` Riemannian metric on a manifold `M`
+    modelled on `(E_ℝ, H_ℝ)` with model `I_ℝ`. This specializes `PseudoRiemannianMetric` to the
+    case where the base field is `ℝ` and requires the bilinear form `gₓ` at each point `x` to be
+    positive-definite.
+*   `RiemannianMetric.inner g x`: The inner product (bilinear form `gₓ`) on the tangent space `TₓM`.
+*   `RiemannianMetric.sharp g x`: The musical isomorphism (index raising) from the cotangent space
+    `T*ₓM` to the tangent space `TₓM`, which is an isomorphism in the Riemannian case due to
+    positive definiteness.
+*   `RiemannianMetric.toQuadraticForm g x`: Expresses the metric at point `x` as a quadratic form.
+
+## Implementation Notes
+
+*   Extends `PseudoRiemannianMetric` by fixing the base field to `ℝ` and adding the `pos_def'`
+    field, ensuring `gₓ(v, v) > 0` for non-zero `v`.
+*   The positive definiteness allows defining an `InnerProductSpace ℝ (TₓM)` instance on each
+    tangent space `TₓM`.
+-/
+universe u v w
+
+open PseudoRiemannianMetric Bundle ContinuousLinearMap
+
+noncomputable section
+
+variable {E_ℝ : Type v} [NormedAddCommGroup E_ℝ] [NormedSpace ℝ E_ℝ] [FiniteDimensional ℝ E_ℝ]
+variable {H_ℝ : Type w} [TopologicalSpace H_ℝ]
+variable {M_ℝ : Type w} [TopologicalSpace M_ℝ] [ChartedSpace H_ℝ M_ℝ] [ChartedSpace H_ℝ E_ℝ]
+
+/-- A Riemannian metric of smoothness class `n` on a manifold `M` over `ℝ`.
+    This extends a pseudo-Riemannian metric with the positive definiteness condition. -/
+@[ext]
+structure RiemannianMetric
+  (I_ℝ : ModelWithCorners ℝ E_ℝ H_ℝ)
+  (n    : WithTop ℕ∞)
+  [inst_top : TopologicalSpace (TangentBundle I_ℝ M_ℝ)]
+  [inst_fib : FiberBundle      E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
+  [inst_vec : VectorBundle    ℝ E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
+  [inst_mani : IsManifold       I_ℝ (n + 1) M_ℝ]
+  [inst_cmvb : ContMDiffVectorBundle n E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _) I_ℝ]
+  extends @PseudoRiemannianMetric ℝ Real.instRCLike E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib
+    inst_vec inst_mani inst_cmvb where
+  /-- `gₓ(v, v) > 0` for all nonzero `v`. -/
+  pos_def' : ∀ x v, v ≠ 0 → val x v v > 0
+
+
+namespace RiemannianMetric
+
+-- Propagate instance assumptions
+variable {I_ℝ : ModelWithCorners ℝ E_ℝ H_ℝ} {n : WithTop ℕ∞}
+variable [inst_top : TopologicalSpace (TangentBundle I_ℝ M_ℝ)]
+variable [inst_fib : FiberBundle E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
+variable [inst_vec : VectorBundle ℝ E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
+variable [inst_mani : IsManifold I_ℝ (n + 1) M_ℝ]
+variable [inst_cmvb : ContMDiffVectorBundle n E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _) I_ℝ]
+
+/-- Coercion from RiemannianMetric to its underlying PseudoRiemannianMetric. -/
+instance : Coe (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
+      inst_cmvb)
+                (@PseudoRiemannianMetric ℝ Real.instRCLike E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top
+         inst_fib inst_vec inst_mani inst_cmvb) where
+  coe g := g.toPseudoRiemannianMetric
+
+omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
+@[simp] lemma pos_def (g : RiemannianMetric I_ℝ n) (x : M_ℝ) (v : TangentSpace I_ℝ x) (hv : v ≠ 0) :
+  (g.toPseudoRiemannianMetric.val x v) v > 0 := g.pos_def' x v hv
+
+/-- The inverse of the musical isomorphism (index raising), which is an isomorphism
+    in the Riemannian case. This is well-defined because the metric is positive definite. -/
+def sharp
+    (g : (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
+      inst_cmvb)) (x : M_ℝ) :
+    Module.Dual ℝ (TangentSpace I_ℝ x) ≃ₗ[ℝ] TangentSpace I_ℝ x :=
+  let bilin := g.toPseudoRiemannianMetric.toBilinForm x
+  have h_nondeg : bilin.Nondegenerate :=
+    g.toPseudoRiemannianMetric.toBilinForm_nondegenerate x
+  LinearEquiv.symm (bilin.toDual h_nondeg)
+
+/-- Express a Riemannian metric at a point as a quadratic form. -/
+abbrev toQuadraticForm
+    (g : (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
+      inst_cmvb))
+    (x : M_ℝ) :
+    QuadraticForm ℝ (TangentSpace I_ℝ x) :=
+  g.toPseudoRiemannianMetric.toQuadraticForm x
+
+omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
+/-- The quadratic form associated with a Riemannian metric is positive definite. -/
+lemma toQuadraticForm_posDef (g : RiemannianMetric I_ℝ n) (x : M_ℝ) :
+    (g.toQuadraticForm x).PosDef :=
+  λ v hv => g.pos_def x v hv
+
+omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
+/-- The application of a Riemannian metric's quadratic form to a vector. -/
+lemma toQuadraticForm_apply (g : RiemannianMetric I_ℝ n) (x : M_ℝ)
+    (v : TangentSpace I_ℝ x) :
+    g.toQuadraticForm x v = g.val x v v := by
+  simp only [toQuadraticForm, PseudoRiemannianMetric.toQuadraticForm_apply]
+
+omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
+/-- A positive definite quadratic form is nondegenerate. -/
+lemma posDef_to_nondegenerate [Invertible (2 : ℝ)]
+    (g : RiemannianMetric I_ℝ n) (x : M_ℝ) :
+    (QuadraticMap.associated (R := ℝ) (N := ℝ) (g.toQuadraticForm x)).Nondegenerate := by
+  let Q := g.toQuadraticForm x
+  let B := QuadraticMap.associated (R := ℝ) (N := ℝ) Q
+  constructor
+  · intro v h_all_zero_w
+    specialize h_all_zero_w v
+    have h_assoc_self : B v v = Q v :=
+        QuadraticMap.associated_eq_self_apply ℝ Q v
+    rw [h_assoc_self] at h_all_zero_w
+    by_cases h_v_zero : v = 0
+    · exact h_v_zero
+    · have h_quad_pos : Q v > 0 := g.pos_def x v h_v_zero
+      linarith [h_all_zero_w, h_quad_pos]
+  · intro w h_all_zero_v
+    specialize h_all_zero_v w
+    have h_assoc_self : B w w = Q w :=
+        QuadraticMap.associated_eq_self_apply ℝ Q w
+    rw [h_assoc_self] at h_all_zero_v
+    by_cases h_w_zero : w = 0
+    · exact h_w_zero
+    · have h_quad_pos : Q w > 0 := g.pos_def x w h_w_zero
+      linarith [h_all_zero_v, h_quad_pos]
+
+/-- The isometry between `(Q.prod <| -Q)` and `QuadraticForm.dualProd`,
+    where `Q` is the quadratic form associated with the Riemannian metric at `x`. -/
+abbrev toDualProdIso [Invertible (2 : ℝ)] (g : RiemannianMetric I_ℝ n) (x : M_ℝ) :
+    ((g.toQuadraticForm x).prod <| -(g.toQuadraticForm x)) →qᵢ
+    (QuadraticForm.dualProd ℝ (TangentSpace I_ℝ x)) :=
+  QuadraticForm.toDualProd (g.toQuadraticForm x)
+
+/-- The inner product on the tangent space at point `x` induced by the Riemannian metric `g`. -/
+def inner
+    (g : (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
+      inst_cmvb)) (x : M_ℝ) (v w : TangentSpace I_ℝ x) : ℝ :=
+  g.toPseudoRiemannianMetric.val x v w
+
+omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
+@[simp] lemma inner_apply (g : RiemannianMetric I_ℝ n) (x : M_ℝ) (v w : TangentSpace I_ℝ x) :
+  inner g x v w = g.val x v w := rfl
+
+variable (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n _ _ _ _ _) (x : M_ℝ)
+
+omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
+/-- Pointwise symmetry of the inner product. -/
+lemma inner_symm (v w : TangentSpace I_ℝ x) :
+    g.inner x v w = g.inner x w v := by
+  simp only [inner_apply]
+  exact g.toPseudoRiemannianMetric.symm' x v w
+
+/-- The inner product space core for the tangent space at a point, derived from the
+Riemannian metric. -/
+noncomputable def tangentInnerCore
+  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
+  (x : M_ℝ) :
+  InnerProductSpace.Core ℝ (TangentSpace I_ℝ x) where
+  inner := λ v w => g.inner x v w
+  conj_symm := λ v w => by
+    simp only [inner_apply, conj_trivial]
+    exact g.toPseudoRiemannianMetric.symm' x w v
+  nonneg_re := λ v => by
+    simp only [inner_apply, RCLike.re_to_real]
+    by_cases hv : v = 0
+    · simp [hv, inner_apply, map_zero, zero_apply]
+    · exact le_of_lt (g.pos_def x v hv)
+  add_left := λ u v w => by
+    simp only [inner_apply, map_add, add_apply]
+  smul_left := λ r u v => by
+    simp only [inner_apply, map_smul, smul_apply, conj_trivial]
+    rfl
+  definite := fun v (h_inner_zero : g.inner x v v = 0) => by
+    by_contra h_v_ne_zero
+    have h_pos : g.inner x v v > 0 := g.pos_def x v h_v_ne_zero
+    linarith [h_inner_zero, h_pos]
+
+/-! ### Inner product space structure. -/
+
+/-- Normed additive commutative group structure on the tangent space at a point `x`,
+derived from the Riemannian metric `g`. -/
+noncomputable def TangentSpace.metricNormedAddCommGroup
+  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
+  (x : M_ℝ) :
+  NormedAddCommGroup (TangentSpace I_ℝ x) :=
+  @InnerProductSpace.Core.toNormedAddCommGroup ℝ (TangentSpace I_ℝ x) _ _ _ (tangentInnerCore g x)
+
+/-- The pre-inner product space core for the tangent space at a point,
+derived from the Riemannian metric. -/
+noncomputable def tangentPreInnerCore
+  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
+  (x : M_ℝ) :
+  PreInnerProductSpace.Core ℝ (TangentSpace I_ℝ x) where
+  inner := λ v w => g.inner x v w
+  conj_symm := λ v w => by
+    simp only [inner_apply, conj_trivial]
+    exact g.toPseudoRiemannianMetric.symm' x w v
+  nonneg_re := λ v => by
+    simp only [inner_apply, RCLike.re_to_real]
+    by_cases hv : v = 0
+    · simp [hv, inner_apply, map_zero, zero_apply]
+    · exact le_of_lt (g.pos_def x v hv)
+  add_left := λ u v w => by
+    simp only [inner_apply, map_add, add_apply]
+  smul_left := λ r u v => by
+    simp only [inner_apply, map_smul, smul_apply, conj_trivial]
+    rfl
+
+/-- Each tangent space carries a seminormed add comm group structure derived from
+the Riemannian metric -/
+noncomputable def TangentSpace.metricSeminormedAddCommGroup
+  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
+  (x : M_ℝ) :
+  SeminormedAddCommGroup (TangentSpace I_ℝ x) :=
+  (TangentSpace.metricNormedAddCommGroup g x).toSeminormedAddCommGroup
diff --git a/PhysLean/Relativity/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Relativity/Metric/PseudoRiemannian/Defs.lean
new file mode 100644
index 000000000..dd81704d1
--- /dev/null
+++ b/PhysLean/Relativity/Metric/PseudoRiemannian/Defs.lean
@@ -0,0 +1,171 @@
+/- Lean 4 code -/
+import Mathlib.Geometry.Manifold.VectorBundle.Tangent
+import Mathlib.Algebra.Module.Basic
+import Mathlib.Geometry.Manifold.Instances.Real
+import Mathlib.Geometry.Manifold.ContMDiffMap
+import Mathlib.Geometry.Manifold.VectorBundle.SmoothSection
+import Mathlib.LinearAlgebra.FiniteDimensional
+import Mathlib.LinearAlgebra.BilinearForm.Properties
+import Mathlib.Topology.Connected.Basic
+import Mathlib.Geometry.Manifold.VectorBundle.Basic
+import Mathlib.Geometry.Manifold.ContMDiff.Basic
+import Mathlib.Analysis.Calculus.ContDiff.Basic
+import Mathlib.Topology.FiberBundle.Basic
+import Mathlib.Algebra.Module.LinearMap.Defs
+import Mathlib.LinearAlgebra.Dual -- Needed for Module.Dual
+import Mathlib.LinearAlgebra.BilinearForm.Basic
+import Mathlib.Geometry.Manifold.VectorBundle.Hom
+import Mathlib.Geometry.Manifold.IsManifold.Basic
+import Mathlib.Analysis.Calculus.ContDiff.FiniteDimension
+import Mathlib.Topology.Algebra.Module.LinearMap
+import Mathlib.Topology.VectorBundle.Constructions -- Added for VectorBundle.dual
+import Mathlib.LinearAlgebra.FiniteDimensional.Defs
+import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
+
+
+--section PseudoRiemannianMetric
+
+universe u v w
+
+open Bundle Set Function Filter Module -- Added Module to open namespaces
+open scoped Manifold Topology Bundle LinearMap.dualMap
+
+abbrev contMDiff_linear {𝕜 : Type u} [NontriviallyNormedField 𝕜]
+    {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+    {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+    (f : E →L[𝕜] F) {n : ℕ∞} :
+    ContMDiff 𝓘(𝕜, E) 𝓘(𝕜, F) n f :=
+  f.contDiff.contMDiff
+
+variable {𝕜 : Type u} [NontriviallyNormedField 𝕜]
+variable {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E]
+variable {H : Type w} [TopologicalSpace H]
+variable {M : Type w} [TopologicalSpace M] [ChartedSpace H M]
+
+/-- A pseudo-Riemannian metric of smoothness class `n` on a manifold `M`.
+    This structure separates the smoothness parameter from the context to allow
+    composition of structures with different smoothness requirements. -/
+@[ext]
+structure PseudoRiemannianMetric
+    {𝕜 : Type u} [NontriviallyNormedField 𝕜]
+    {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+    {H : Type w} [TopologicalSpace H]
+    {M : Type w} [TopologicalSpace M]
+    [ChartedSpace H M] -- Removed [ChartedSpace H E] as it seemed incorrect
+    (I : ModelWithCorners 𝕜 E H)
+    (n : WithTop ℕ∞)
+    -- Now list instances depending on I and n
+    -- Ensure necessary instances for TangentBundle are available
+    -- These instances are typically inferred when M is a manifold.
+    [TopologicalSpace (TangentBundle I M)]
+    [FiberBundle E (TangentSpace I : M → Type _)] -- Use explicit type family
+    [VectorBundle 𝕜 E (TangentSpace I : M → Type _)] -- Use explicit type family
+    -- Add IsManifold and ContMDiffVectorBundle constraints here
+    -- Require IsManifold I (n + 1) M to ensure ContMDiffVectorBundle n can be inferred
+    [IsManifold I (n + 1) M] -- Removed name h_manifold
+    [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I] -- Removed name h_contMDiffVec
+   : Type (max u v w) where
+  /-- The metric as a function on tangent vectors, providing a bilinear form at each point. -/
+  val : ∀ (x : M), LinearMap.BilinForm 𝕜 (TangentSpace I x)
+  /-- The metric is symmetric: g(v,w) = g(w,v) for all tangent vectors v, w. -/
+  symm : ∀ (x : M), (val x).IsSymm
+  /-- The metric is non-degenerate: if g(v,w) = 0 for all w, then v = 0. -/
+  nondegenerate : ∀ (x : M), (val x).Nondegenerate
+  /-- The metric is smooth of class `n`: it varies smoothly across the manifold when applied to
+      smooth vector fields. -/
+  smooth : ∀ (X Y : Cₛ^n⟮I; E, (TangentSpace I : M → Type _)⟯), -- Use explicit type family
+    ContMDiff I (modelWithCornersSelf 𝕜 𝕜) n (λ x => val x (X x) (Y x))
+
+namespace PseudoRiemannianMetric
+
+--variable [FiniteDimensional 𝕜 E]
+
+-- Define the model fiber for bilinear forms
+@[nolint unusedArguments]
+abbrev ModelBilinForm (_𝕜 E : Type*) [NontriviallyNormedField _𝕜] [NormedAddCommGroup E] [NormedSpace _𝕜 E] : Type _ :=
+  E →L[_𝕜] (E →L[_𝕜] _𝕜)
+
+-- Define the bundle of continuous bilinear forms on the tangent bundle
+-- The fiber at x is the space of continuous linear maps from TangentSpace I x to its continuous dual.
+@[nolint unusedArguments]
+def TangentBilinFormBundle (𝕜 : Type u) [NontriviallyNormedField 𝕜]
+    {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+    {H : Type w} [TopologicalSpace H]
+    {M : Type w} [TopologicalSpace M] [ChartedSpace H M]
+    (I : ModelWithCorners 𝕜 E H)
+    -- Instances ensuring TangentSpace I x is a normed space and the bundle structure exists
+    [TopologicalSpace (TangentBundle I M)]
+    [FiberBundle E (TangentSpace I : M → Type _)]
+    [VectorBundle 𝕜 E (TangentSpace I : M → Type _)] : M → Type _ :=
+  fun x => TangentSpace I x →L[𝕜] (TangentSpace I x →L[𝕜] 𝕜)
+
+/-- The fibers of a vector bundle are finite-dimensional if the model fiber is finite-dimensional. -/
+noncomputable def VectorBundle.finiteDimensional_fiber
+    (𝕜 : Type u) [NontriviallyNormedField 𝕜]
+    {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F]
+    {B : Type w} [TopologicalSpace B]
+    {E : B → Type*} [∀ x, AddCommGroup (E x)] [∀ x, Module 𝕜 (E x)]
+    [TopologicalSpace (Bundle.TotalSpace F E)]
+    [∀ x, TopologicalSpace (E x)]
+    [FiberBundle F E]
+    [VectorBundle 𝕜 F E]
+    (x : B) :
+    FiniteDimensional 𝕜 (E x) :=
+  -- Get a trivialization at `x` from the FiberBundle atlas
+  let triv := trivializationAt F E x
+  -- Get the fact that x is in the base set
+  let hx := mem_baseSet_trivializationAt F E x
+  -- Obtain linear property through the vector bundle structure
+  have h_linear : triv.IsLinear 𝕜 := by
+    apply VectorBundle.trivialization_linear'
+  -- Register linearity as an instance
+  haveI : triv.IsLinear 𝕜 := h_linear
+  -- Get the linear equivalence between the fiber E x and the model fiber F
+  let linear_equiv := triv.continuousLinearEquivAt 𝕜 x hx
+  -- Transfer the finite-dimensionality from F to E x via the linear equivalence
+  have : FiniteDimensional 𝕜 (E x) :=
+    LinearEquiv.finiteDimensional linear_equiv.symm.toLinearEquiv
+  this
+
+-- Instance providing FiniteDimensional for tangent spaces, needed for VectorBundle.dual
+-- Require the VectorBundle instance explicitly as an argument
+noncomputable instance TangentSpace.finiteDimensional
+  (𝕜 : Type u) [NontriviallyNormedField 𝕜]
+  {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] -- Model Fiber E
+  {H : Type w} [TopologicalSpace H]
+  {M : Type w} [TopologicalSpace M] [ChartedSpace H M]
+  (I : ModelWithCorners 𝕜 E H)
+  -- Add SmoothManifoldWithCorners instance, which implies the bundle structures
+  [IsManifold I ⊤ M] -- Specify smoothness level, e.g., C^infinity
+  (x : M) :
+  FiniteDimensional 𝕜 (TangentSpace I x) :=
+  -- The fibers of a vector bundle with a finite-dimensional model fiber are finite-dimensional.
+  -- The FiberBundle and VectorBundle instances are inferred from IsManifold I ⊤ M.
+  VectorBundle.finiteDimensional_fiber 𝕜 (F := E) (E := TangentSpace I) x
+
+-- Instance for the dual bundle (VectorBundle)
+section DualBundle
+  variable (𝕜 : Type u) [NontriviallyNormedField 𝕜]
+    {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E]
+    {H : Type w} [TopologicalSpace H]
+    {M : Type w} [TopologicalSpace M] [ChartedSpace H M] -- Explicitly require ChartedSpace
+    (I : ModelWithCorners 𝕜 E H)
+    [IsManifold I ⊤ M] -- This requires ChartedSpace H M and provides TangentBundle structure
+
+  -- Define the dual bundle type family
+  abbrev TangentDualFiber (x : M) := TangentSpace I x →L[𝕜] 𝕜
+  -- Define the model fiber for the dual bundle
+  abbrev TangentDualModelFiber := E →L[𝕜] 𝕜
+
+  -- Explicitly provide instances for the dual model fiber
+  noncomputable instance : NormedAddCommGroup (E →L[𝕜] 𝕜) := inferInstance
+  noncomputable instance : NormedSpace 𝕜 (E →L[𝕜] 𝕜) := inferInstance
+  -- Finite dimensionality of the dual model fiber requires FiniteDimensional 𝕜 E
+  noncomputable instance : FiniteDimensional 𝕜 (E →L[𝕜] 𝕜) := inferInstance
+
+-- Instance for the trivial vector bundle M × 𝕜 needed for continuousLinearMap
+  noncomputable instance TrivialBundleInstance : VectorBundle 𝕜 𝕜 (fun _ : M => 𝕜) :=
+    Bundle.Trivial.vectorBundle 𝕜 M 𝕜
+
+end DualBundle
+end PseudoRiemannianMetric

From cd65b2500e13d09c607dde9ee994888162d955b5 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Thu, 24 Apr 2025 11:00:18 +0200
Subject: [PATCH 08/32] Delete Defs.lean

---
 .../Metric/PseudoRiemannian/Defs.lean         | 171 ------------------
 1 file changed, 171 deletions(-)
 delete mode 100644 PhysLean/Relativity/Metric/PseudoRiemannian/Defs.lean

diff --git a/PhysLean/Relativity/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Relativity/Metric/PseudoRiemannian/Defs.lean
deleted file mode 100644
index dd81704d1..000000000
--- a/PhysLean/Relativity/Metric/PseudoRiemannian/Defs.lean
+++ /dev/null
@@ -1,171 +0,0 @@
-/- Lean 4 code -/
-import Mathlib.Geometry.Manifold.VectorBundle.Tangent
-import Mathlib.Algebra.Module.Basic
-import Mathlib.Geometry.Manifold.Instances.Real
-import Mathlib.Geometry.Manifold.ContMDiffMap
-import Mathlib.Geometry.Manifold.VectorBundle.SmoothSection
-import Mathlib.LinearAlgebra.FiniteDimensional
-import Mathlib.LinearAlgebra.BilinearForm.Properties
-import Mathlib.Topology.Connected.Basic
-import Mathlib.Geometry.Manifold.VectorBundle.Basic
-import Mathlib.Geometry.Manifold.ContMDiff.Basic
-import Mathlib.Analysis.Calculus.ContDiff.Basic
-import Mathlib.Topology.FiberBundle.Basic
-import Mathlib.Algebra.Module.LinearMap.Defs
-import Mathlib.LinearAlgebra.Dual -- Needed for Module.Dual
-import Mathlib.LinearAlgebra.BilinearForm.Basic
-import Mathlib.Geometry.Manifold.VectorBundle.Hom
-import Mathlib.Geometry.Manifold.IsManifold.Basic
-import Mathlib.Analysis.Calculus.ContDiff.FiniteDimension
-import Mathlib.Topology.Algebra.Module.LinearMap
-import Mathlib.Topology.VectorBundle.Constructions -- Added for VectorBundle.dual
-import Mathlib.LinearAlgebra.FiniteDimensional.Defs
-import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
-
-
---section PseudoRiemannianMetric
-
-universe u v w
-
-open Bundle Set Function Filter Module -- Added Module to open namespaces
-open scoped Manifold Topology Bundle LinearMap.dualMap
-
-abbrev contMDiff_linear {𝕜 : Type u} [NontriviallyNormedField 𝕜]
-    {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-    {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
-    (f : E →L[𝕜] F) {n : ℕ∞} :
-    ContMDiff 𝓘(𝕜, E) 𝓘(𝕜, F) n f :=
-  f.contDiff.contMDiff
-
-variable {𝕜 : Type u} [NontriviallyNormedField 𝕜]
-variable {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E]
-variable {H : Type w} [TopologicalSpace H]
-variable {M : Type w} [TopologicalSpace M] [ChartedSpace H M]
-
-/-- A pseudo-Riemannian metric of smoothness class `n` on a manifold `M`.
-    This structure separates the smoothness parameter from the context to allow
-    composition of structures with different smoothness requirements. -/
-@[ext]
-structure PseudoRiemannianMetric
-    {𝕜 : Type u} [NontriviallyNormedField 𝕜]
-    {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-    {H : Type w} [TopologicalSpace H]
-    {M : Type w} [TopologicalSpace M]
-    [ChartedSpace H M] -- Removed [ChartedSpace H E] as it seemed incorrect
-    (I : ModelWithCorners 𝕜 E H)
-    (n : WithTop ℕ∞)
-    -- Now list instances depending on I and n
-    -- Ensure necessary instances for TangentBundle are available
-    -- These instances are typically inferred when M is a manifold.
-    [TopologicalSpace (TangentBundle I M)]
-    [FiberBundle E (TangentSpace I : M → Type _)] -- Use explicit type family
-    [VectorBundle 𝕜 E (TangentSpace I : M → Type _)] -- Use explicit type family
-    -- Add IsManifold and ContMDiffVectorBundle constraints here
-    -- Require IsManifold I (n + 1) M to ensure ContMDiffVectorBundle n can be inferred
-    [IsManifold I (n + 1) M] -- Removed name h_manifold
-    [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I] -- Removed name h_contMDiffVec
-   : Type (max u v w) where
-  /-- The metric as a function on tangent vectors, providing a bilinear form at each point. -/
-  val : ∀ (x : M), LinearMap.BilinForm 𝕜 (TangentSpace I x)
-  /-- The metric is symmetric: g(v,w) = g(w,v) for all tangent vectors v, w. -/
-  symm : ∀ (x : M), (val x).IsSymm
-  /-- The metric is non-degenerate: if g(v,w) = 0 for all w, then v = 0. -/
-  nondegenerate : ∀ (x : M), (val x).Nondegenerate
-  /-- The metric is smooth of class `n`: it varies smoothly across the manifold when applied to
-      smooth vector fields. -/
-  smooth : ∀ (X Y : Cₛ^n⟮I; E, (TangentSpace I : M → Type _)⟯), -- Use explicit type family
-    ContMDiff I (modelWithCornersSelf 𝕜 𝕜) n (λ x => val x (X x) (Y x))
-
-namespace PseudoRiemannianMetric
-
---variable [FiniteDimensional 𝕜 E]
-
--- Define the model fiber for bilinear forms
-@[nolint unusedArguments]
-abbrev ModelBilinForm (_𝕜 E : Type*) [NontriviallyNormedField _𝕜] [NormedAddCommGroup E] [NormedSpace _𝕜 E] : Type _ :=
-  E →L[_𝕜] (E →L[_𝕜] _𝕜)
-
--- Define the bundle of continuous bilinear forms on the tangent bundle
--- The fiber at x is the space of continuous linear maps from TangentSpace I x to its continuous dual.
-@[nolint unusedArguments]
-def TangentBilinFormBundle (𝕜 : Type u) [NontriviallyNormedField 𝕜]
-    {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-    {H : Type w} [TopologicalSpace H]
-    {M : Type w} [TopologicalSpace M] [ChartedSpace H M]
-    (I : ModelWithCorners 𝕜 E H)
-    -- Instances ensuring TangentSpace I x is a normed space and the bundle structure exists
-    [TopologicalSpace (TangentBundle I M)]
-    [FiberBundle E (TangentSpace I : M → Type _)]
-    [VectorBundle 𝕜 E (TangentSpace I : M → Type _)] : M → Type _ :=
-  fun x => TangentSpace I x →L[𝕜] (TangentSpace I x →L[𝕜] 𝕜)
-
-/-- The fibers of a vector bundle are finite-dimensional if the model fiber is finite-dimensional. -/
-noncomputable def VectorBundle.finiteDimensional_fiber
-    (𝕜 : Type u) [NontriviallyNormedField 𝕜]
-    {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F]
-    {B : Type w} [TopologicalSpace B]
-    {E : B → Type*} [∀ x, AddCommGroup (E x)] [∀ x, Module 𝕜 (E x)]
-    [TopologicalSpace (Bundle.TotalSpace F E)]
-    [∀ x, TopologicalSpace (E x)]
-    [FiberBundle F E]
-    [VectorBundle 𝕜 F E]
-    (x : B) :
-    FiniteDimensional 𝕜 (E x) :=
-  -- Get a trivialization at `x` from the FiberBundle atlas
-  let triv := trivializationAt F E x
-  -- Get the fact that x is in the base set
-  let hx := mem_baseSet_trivializationAt F E x
-  -- Obtain linear property through the vector bundle structure
-  have h_linear : triv.IsLinear 𝕜 := by
-    apply VectorBundle.trivialization_linear'
-  -- Register linearity as an instance
-  haveI : triv.IsLinear 𝕜 := h_linear
-  -- Get the linear equivalence between the fiber E x and the model fiber F
-  let linear_equiv := triv.continuousLinearEquivAt 𝕜 x hx
-  -- Transfer the finite-dimensionality from F to E x via the linear equivalence
-  have : FiniteDimensional 𝕜 (E x) :=
-    LinearEquiv.finiteDimensional linear_equiv.symm.toLinearEquiv
-  this
-
--- Instance providing FiniteDimensional for tangent spaces, needed for VectorBundle.dual
--- Require the VectorBundle instance explicitly as an argument
-noncomputable instance TangentSpace.finiteDimensional
-  (𝕜 : Type u) [NontriviallyNormedField 𝕜]
-  {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] -- Model Fiber E
-  {H : Type w} [TopologicalSpace H]
-  {M : Type w} [TopologicalSpace M] [ChartedSpace H M]
-  (I : ModelWithCorners 𝕜 E H)
-  -- Add SmoothManifoldWithCorners instance, which implies the bundle structures
-  [IsManifold I ⊤ M] -- Specify smoothness level, e.g., C^infinity
-  (x : M) :
-  FiniteDimensional 𝕜 (TangentSpace I x) :=
-  -- The fibers of a vector bundle with a finite-dimensional model fiber are finite-dimensional.
-  -- The FiberBundle and VectorBundle instances are inferred from IsManifold I ⊤ M.
-  VectorBundle.finiteDimensional_fiber 𝕜 (F := E) (E := TangentSpace I) x
-
--- Instance for the dual bundle (VectorBundle)
-section DualBundle
-  variable (𝕜 : Type u) [NontriviallyNormedField 𝕜]
-    {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E]
-    {H : Type w} [TopologicalSpace H]
-    {M : Type w} [TopologicalSpace M] [ChartedSpace H M] -- Explicitly require ChartedSpace
-    (I : ModelWithCorners 𝕜 E H)
-    [IsManifold I ⊤ M] -- This requires ChartedSpace H M and provides TangentBundle structure
-
-  -- Define the dual bundle type family
-  abbrev TangentDualFiber (x : M) := TangentSpace I x →L[𝕜] 𝕜
-  -- Define the model fiber for the dual bundle
-  abbrev TangentDualModelFiber := E →L[𝕜] 𝕜
-
-  -- Explicitly provide instances for the dual model fiber
-  noncomputable instance : NormedAddCommGroup (E →L[𝕜] 𝕜) := inferInstance
-  noncomputable instance : NormedSpace 𝕜 (E →L[𝕜] 𝕜) := inferInstance
-  -- Finite dimensionality of the dual model fiber requires FiniteDimensional 𝕜 E
-  noncomputable instance : FiniteDimensional 𝕜 (E →L[𝕜] 𝕜) := inferInstance
-
--- Instance for the trivial vector bundle M × 𝕜 needed for continuousLinearMap
-  noncomputable instance TrivialBundleInstance : VectorBundle 𝕜 𝕜 (fun _ : M => 𝕜) :=
-    Bundle.Trivial.vectorBundle 𝕜 M 𝕜
-
-end DualBundle
-end PseudoRiemannianMetric

From e603d8e7f5c14a71d102851752186e6d5bd827ce Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Tue, 29 Apr 2025 01:33:03 +0200
Subject: [PATCH 09/32] Update PseudoRiemannian and Riemannian

smoothness is now defined locally as a predicate
added constant index condition (cfr O'Neill 'Semi-Riemannian Geometry p 54,55)
Removed RCLike as it should not be needed for general relativity or QFT
InnerProductSpace is defined locally instead of as a global instance
---
 .../Metric/PseudoRiemannian/Defs.lean         | 604 +++++++++---------
 .../Geometry/Metric/Riemannian/Defs.lean      | 281 ++++----
 .../Geometry/Metric/Riemannian/DefsOrig.lean  | 235 +++++++
 3 files changed, 666 insertions(+), 454 deletions(-)
 create mode 100644 PhysLean/Mathematics/Geometry/Metric/Riemannian/DefsOrig.lean

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index c82f74683..9ecbb1b6c 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -5,25 +5,31 @@ Authors: Matteo Cipollina
 -/
 
 import Mathlib.Analysis.InnerProductSpace.Basic
-import Mathlib.Geometry.Manifold.VectorBundle.SmoothSection
+import Mathlib.Analysis.RCLike.Lemmas
+import Mathlib.Geometry.Manifold.MFDeriv.Defs
+import Mathlib.Geometry.Manifold.VectorBundle.Basic
 import Mathlib.LinearAlgebra.BilinearForm.Properties
-import Mathlib.LinearAlgebra.QuadraticForm.Basic
+import Mathlib.LinearAlgebra.QuadraticForm.Real
+import Mathlib.Topology.LocallyConstant.Basic
 
 /-!
 # Pseudo-Riemannian Metrics on Smooth Manifolds
 
-This file formalizes the fundamental notion of pseudo-Riemannian metrics on smooth manifolds
-and establishes their basic properties. A pseudo-Riemannian metric equips a manifold with a
-smoothly varying, non-degenerate, symmetric bilinear form on each tangent space, generalizing
-the concept of an inner product space to curved spaces.
+This file formalizes pseudo-Riemannian metrics on smooth manifolds and establishes their basic
+properties.
+
+A pseudo-Riemannian metric equips a manifold with a smoothly varying, non-degenerate, symmetric
+bilinear form of *constant index* on each tangent space, generalizing the concept of an inner
+product space to curved spaces. The index here refers to `QuadraticForm.negDim`, the dimension
+of a maximal negative definite subspace.
 
 ## Main Definitions
 
-* `PseudoRiemannianMetric I n M`: A structure representing a `C^n` pseudo-Riemannian metric
+* `PseudoRiemannianMetric E H M n I`: A structure representing a `C^n` pseudo-Riemannian metric
   on a manifold `M` modelled on `(E, H)` with model with corners `I`. It consists of a family
   of non-degenerate, symmetric, continuous bilinear forms `gₓ` on each tangent space `TₓM`,
-  varying `C^n`-smoothly with `x`. The base field `𝕜` is required to be `RCLike`, the model
-  space `E` must be finite-dimensional, and the manifold `M` must be `C^{n+1}` smooth.
+  varying `C^n`-smoothly with `x` and having a locally constant negative dimension (`negDim`).
+  The model space `E` must be finite-dimensional, and the manifold `M` must be `C^{n+1}` smooth.
 
 * `PseudoRiemannianMetric.flatEquiv g x`: The "musical isomorphism" from the tangent space at `x`
   to its dual space, representing the canonical isomorphism induced by the metric.
@@ -31,21 +37,8 @@ the concept of an inner product space to curved spaces.
 * `PseudoRiemannianMetric.sharpEquiv g x`: The inverse of the flat isomorphism, mapping from
   the dual space back to the tangent space.
 
-## Implementation Notes
-
-* The bilinear form `gₓ` at a point `x` is represented as a `ContinuousLinearMap`
-  `val x : TₓM →L[𝕜] (TₓM →L[𝕜] 𝕜)`. This curried form `v ↦ (w ↦ gₓ(v, w))` encodes both
-  the bilinear structure and its continuity.
-
-* Smoothness is defined by requiring that for any two `C^n` vector fields `X, Y`, the scalar
-  function `x ↦ gₓ(X x, Y x)` is `C^n` smooth (`ContMDiff n`).
-
-* The manifold `M` must be `C^{n+1}` smooth (`IsManifold I (n + 1) M`) to ensure the tangent
-  bundle is a `C^n` vector bundle, making the notion of `C^n` vector fields well-defined.
-
-* Finite-dimensionality of the model space (`FiniteDimensional 𝕜 E`) guarantees that tangent
-  spaces are finite-dimensional, which is necessary for establishing that non-degeneracy implies
-  the existence of an isomorphism between the tangent space and its dual.
+* `PseudoRiemannianMetric.toQuadraticForm g x`: The quadratic form `v ↦ gₓ(v, v)` associated
+  with the metric at point `x`.
 -/
 
 section PseudoRiemannianMetric
@@ -54,267 +47,325 @@ noncomputable section
 
 universe u v w
 
-open Bundle Set Function Filter Module Topology ContinuousLinearMap
+open Bundle Set Finset Function Filter Module Topology ContinuousLinearMap
 open scoped Manifold Bundle LinearMap Dual
 
-/-- Convert a continuous linear map representing a bilinear form to a `LinearMap.BilinForm`. -/
-def ContinuousLinearMap.toBilinForm
-    {𝕜 E : Type*} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E]
-    (f : E →L[𝕜] (E →L[𝕜] 𝕜)) : LinearMap.BilinForm 𝕜 E :=
-  LinearMap.mk₂ 𝕜 (fun v w => (f v) w)
-    (fun v₁ v₂ w => by dsimp; rw [f.map_add]; rfl)
-    (fun a v w => by dsimp; rw [f.map_smul]; rfl)
-    (fun v w₁ w₂ => by dsimp; rw [(f v).map_add];)
-    (fun a v w => (f v).map_smul a w)
-
--- We use Real-like fields for InnerProductSpace in Riemannian Metric
-variable {𝕜 : Type u} [RCLike 𝕜] -- Stronger assumption, implies NontriviallyNormedField
-variable {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E]
-variable {H : Type w} [TopologicalSpace H] -- Chart space
--- Manifold M and ChartedSpace for E
-variable {M : Type w} [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E]
-variable {I : ModelWithCorners 𝕜 E H} {n : WithTop ℕ∞} -- Model with corners and smoothness level
+namespace QuadraticForm
+
+variable {K : Type*} [Field K] [LinearOrder K]
+
+/-- The negative dimension (or index) of a quadratic form is the dimension
+    of a maximal negative definite subspace. -/
+noncomputable def negDim {E : Type*} [AddCommGroup E]
+    [Module ℝ E] [FiniteDimensional ℝ E]
+    (q : QuadraticForm ℝ E) : ℕ :=
+  by
+    classical
+    let P : (Fin (finrank ℝ E) → SignType) → Prop := fun w =>
+        QuadraticMap.Equivalent q (QuadraticMap.weightedSumSquares ℝ fun i => (w i : ℝ))
+    let h_exists : ∃ w, P w := QuadraticForm.equivalent_signType_weighted_sum_squared q
+    let w := Classical.choose h_exists
+    exact Finset.card (Finset.filter (fun i => w i = SignType.neg) Finset.univ)
+
+/-- The i-th standard basis vector has a 1 in the i-th position. -/
+lemma Pi.basisFun_apply_same {𝕜 ι : Type*} [Field 𝕜] [Fintype ι] [DecidableEq ι] (i : ι) :
+  (Pi.basisFun 𝕜 ι) i i = 1 := by
+  simp only [Pi.basisFun_apply, Pi.single_eq_same]
+
+/-- The i-th standard basis vector has 0 in all positions j ≠ i. -/
+lemma Pi.basisFun_apply_ne {𝕜 ι : Type*} [Field 𝕜] [Fintype ι] [DecidableEq ι] (i j : ι) (h : j ≠ i) :
+  (Pi.basisFun 𝕜 ι) i j = 0 := by
+  simp only [Pi.basisFun_apply, Pi.single_eq_of_ne h]
+
+/-- For a standard basis vector in a weighted sum of squares, only one term in the sum is nonzero. -/
+lemma QuadraticMap.weightedSumSquares_basis_vector {E : Type*} [AddCommGroup E]
+    [Module ℝ E] {weights : Fin (finrank ℝ E) → ℝ}
+    {i : Fin (finrank ℝ E)} (v : Fin (finrank ℝ E) → ℝ)
+    (hv : ∀ j, v j = if j = i then 1 else 0) :
+    QuadraticMap.weightedSumSquares ℝ weights v = weights i := by
+  simp only [QuadraticMap.weightedSumSquares_apply]
+  rw [Finset.sum_eq_single i]
+  · simp only [hv i, ↓reduceIte, mul_one, smul_eq_mul]
+  · intro j _ hj
+    simp only [hv j, if_neg hj, mul_zero, smul_eq_mul]
+  · simp only [Finset.mem_univ, not_true_eq_false, smul_eq_mul, mul_eq_zero, or_self,
+    IsEmpty.forall_iff]
+
+/-- When a quadratic form is equivalent to a weighted sum of squares,
+    negative weights correspond to vectors where the form takes negative values. -/
+lemma neg_weight_implies_neg_value {E : Type*} [AddCommGroup E] [Module ℝ E]
+    {q : QuadraticForm ℝ E} {w : Fin (finrank ℝ E) → SignType}
+    (h_equiv : QuadraticMap.Equivalent q (QuadraticMap.weightedSumSquares ℝ fun i => (w i : ℝ)))
+    {i : Fin (finrank ℝ E)} (hi : w i = SignType.neg) :
+    ∃ v : E, v ≠ 0 ∧ q v < 0 := by
+  let f := Classical.choice h_equiv
+  let v_std : Fin (finrank ℝ E) → ℝ := fun j => if j = i then 1 else 0
+  let v := f.symm v_std
+  have hv_ne_zero : v ≠ 0 := by
+    intro h
+    have : f v = f 0 := by rw [h]
+    have : f (f.symm v_std) = f 0 := by rw [← this]
+    have : v_std = 0 := by
+      rw [← f.apply_symm_apply v_std]
+      exact Eq.trans this (map_zero f)
+    have : v_std i = 0 := by rw [this]; rfl
+    simp only [↓reduceIte, one_ne_zero, v_std] at this
+  have hq_neg : q v < 0 := by
+    have heq : q v = QuadraticMap.weightedSumSquares ℝ (fun j => (w j : ℝ)) v_std :=
+      QuadraticMap.IsometryEquiv.map_app f.symm v_std
+    have hw : QuadraticMap.weightedSumSquares ℝ (fun j => (w j : ℝ)) v_std = (w i : ℝ) := by
+      apply QuadraticMap.weightedSumSquares_basis_vector v_std
+      intro j; simp only [v_std]
+    rw [heq, hw]
+    have : (w i : ℝ) = -1 := by simp only [hi, SignType.neg_eq_neg_one, SignType.coe_neg,
+      SignType.coe_one, v_std]
+    rw [this]
+    exact neg_one_lt_zero
+  exact ⟨v, hv_ne_zero, hq_neg⟩
+
+/-- A positive definite quadratic form cannot have any negative weights
+    in its diagonal representation. -/
+lemma posDef_no_neg_weights {E : Type*} [AddCommGroup E] [Module ℝ E]
+    {q : QuadraticForm ℝ E} (hq : q.PosDef)
+    {w : Fin (finrank ℝ E) → SignType}
+    (h_equiv : QuadraticMap.Equivalent q (QuadraticMap.weightedSumSquares ℝ fun i => (w i : ℝ))) :
+    ∀ i, w i ≠ SignType.neg := by
+  intro i hi
+  obtain ⟨v, hv_ne_zero, hq_neg⟩ := QuadraticForm.neg_weight_implies_neg_value h_equiv hi
+  have hq_pos : 0 < q v := hq v hv_ne_zero
+  exact lt_asymm hq_neg hq_pos
+
+/-- For a positive definite quadratic form, the negative dimension (index) is zero. -/
+theorem rankNeg_eq_zero {E : Type*} [AddCommGroup E]
+    [Module ℝ E] [FiniteDimensional ℝ E] {q : QuadraticForm ℝ E} (hq : q.PosDef) :
+    q.negDim = 0 := by
+  haveI : Invertible (2 : ℝ) := inferInstance
+  unfold QuadraticForm.negDim
+  have h_exists := equivalent_signType_weighted_sum_squared q
+  let w := Classical.choose h_exists
+  have h_equiv : QuadraticMap.Equivalent q (QuadraticMap.weightedSumSquares ℝ fun i => (w i : ℝ)) :=
+    Classical.choose_spec h_exists
+  have h_no_neg : ∀ i, w i ≠ SignType.neg :=
+    QuadraticForm.posDef_no_neg_weights hq h_equiv
+  simp [Finset.card_eq_zero, Finset.filter_eq_empty_iff, h_no_neg]
+  exact fun ⦃x⦄ => h_no_neg x
+
+end QuadraticForm
+
+/-- Helper function to convert the metric tensor at `x` to a quadratic form. -/
+private def PseudoRiemannianMetricValToQuadraticForm
+    {E : Type v} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {H : Type w} [TopologicalSpace H]
+    {M : Type w} [TopologicalSpace M] [ChartedSpace H M]
+    {I : ModelWithCorners ℝ E H}
+    (val : ∀ (x : M), TangentSpace I x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ))
+    (symm : ∀ (x : M) (v w : TangentSpace I x), (val x v) w = (val x w) v)
+    (x : M) : QuadraticForm ℝ (TangentSpace I x) where
+  toFun v := val x v v
+  toFun_smul a v := by
+    simp only [ContinuousLinearMap.map_smul, ContinuousLinearMap.smul_apply, smul_smul, pow_two]
+  exists_companion' :=
+      ⟨ LinearMap.mk₂ ℝ (fun v y => val x v y + val x y v)
+        (fun v₁ v₂ y => by simp only [map_add, add_apply]; ring)
+        (fun a v y => by simp only [map_smul, smul_apply]; ring_nf; exact Eq.symm (smul_add ..))
+        (fun v y₁ y₂ => by simp only [map_add, add_apply]; ring)
+        (fun a v y => by simp only [map_smul, smul_apply]; ring_nf; exact Eq.symm (smul_add ..)),
+            by
+              intro v y
+              simp only [LinearMap.mk₂_apply, ContinuousLinearMap.map_add,
+                ContinuousLinearMap.add_apply, symm x]
+              ring⟩
 
 /-- A pseudo-Riemannian metric of smoothness class `C^n` on a manifold `M` modelled on `(E, H)`
     with model `I`. This structure defines a smoothly varying, non-degenerate, symmetric,
-    continuous bilinear form `gₓ` on the tangent space `TₓM` at each point `x`.
-    Requires `M` to be `C^{n+1}` smooth.
--/
+    continuous bilinear form `gₓ` of constant negative dimension on the tangent space `TₓM`
+    at each point `x`. Requires `M` to be `C^{n+1}` smooth.-/
 @[ext]
 structure PseudoRiemannianMetric
-    (I : ModelWithCorners 𝕜 E H) (n : WithTop ℕ∞)
-    -- Instances ensuring the tangent bundle is a well-defined smooth vector bundle
-    [TopologicalSpace (TangentBundle I M)]
-    [FiberBundle E (TangentSpace I : M → Type _)]
-    [VectorBundle 𝕜 E (TangentSpace I : M → Type _)]
-    [IsManifold I (n + 1) M] -- Manifold is C^{n+1}
-    -- Tangent bundle is C^n
-    [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I] : Type (max u v w) where
+    (E : Type v) (H : Type w) (M : Type w) (n : WithTop ℕ∞)
+    [inst_norm_grp_E : NormedAddCommGroup E]
+    [inst_norm_sp_E : NormedSpace ℝ E]
+    [inst_findim_E : FiniteDimensional ℝ E]
+    [inst_top_H : TopologicalSpace H]
+    [inst_top_M : TopologicalSpace M]
+    [inst_chart_M : ChartedSpace H M]
+    [inst_chart_E : ChartedSpace H E]
+    (I : ModelWithCorners ℝ E H)
+    [inst_mani : IsManifold I (n + 1) M]
+    [inst_total_top : TopologicalSpace (TotalSpace E (TangentSpace I : M → Type _))]
+    [inst_fb : FiberBundle E (TangentSpace I : M → Type _)]
+    [inst_vb : VectorBundle ℝ E (TangentSpace I : M → Type _)]
+    [inst_cmvb : ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
+    [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)] :
+      Type (max u v w) where
   /-- The metric tensor at each point `x : M`, represented as a continuous linear map
-      `TₓM →L[𝕜] (TₓM →L[𝕜] 𝕜)`. Applying it twice, `(val x v) w`, yields `gₓ(v, w)`. -/
-  protected val : ∀ (x : M), TangentSpace I x →L[𝕜] (TangentSpace I x →L[𝕜] 𝕜)
+      `TₓM →L[ℝ] (TₓM →L[ℝ] ℝ)`. Applying it twice, `(val x v) w`, yields `gₓ(v, w)`. -/
+  protected val : ∀ (x : M), TangentSpace I x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ)
   /-- The metric is symmetric: `gₓ(v, w) = gₓ(w, v)`. -/
   protected symm : ∀ (x : M) (v w : TangentSpace I x), (val x v) w = (val x w) v
   /-- The metric is non-degenerate: if `gₓ(v, w) = 0` for all `w`, then `v = 0`. -/
   protected nondegenerate : ∀ (x : M) (v : TangentSpace I x), (∀ w : TangentSpace I x,
     (val x v) w = 0) → v = 0
-  /-- The metric varies smoothly: `x ↦ gₓ(Xₓ, Yₓ)` is a `C^n` function for `C^n`
-      vector fields `X, Y`. -/
-  protected smooth : ∀ (X Y : ContMDiffSection I E n (TangentSpace I)),
-    ContMDiff I (modelWithCornersSelf 𝕜 𝕜) n (fun x => val x (X x) (Y x))
+  /-- The metric varies smoothly: Expressed in local coordinates via any chart `e`, the function
+      `y ↦ g_{e.symm y}(mfderiv I I e.symm y v, mfderiv I I e.symm y w)` is `C^n` smooth on the
+      chart's target `e.target` for any constant vectors `v, w` in the model space `E`. -/
+  protected smooth_in_charts' : ∀ (x₀ : M) (v w : E),
+    let e := extChartAt I x₀
+    ContDiffWithinAt ℝ n
+      (fun y => val (e.symm y) (mfderiv I I e.symm y v) (mfderiv I I e.symm y w))
+      (e.target) (e x₀)
+  /-- The negative dimension (`QuadraticForm.negDim`) of the metric's quadratic form is
+      locally constant. On a connected manifold, this implies it is constant globally. -/
+  protected negDim_isLocallyConstant :
+    IsLocallyConstant (fun x : M =>
+      have : FiniteDimensional ℝ (TangentSpace I x) := inferInstance
+      (PseudoRiemannianMetricValToQuadraticForm val symm x).negDim)
 
 namespace PseudoRiemannianMetric
 
-instance TangentSpace.addCommGroup (x : M) : AddCommGroup (TangentSpace I x) := by infer_instance
-instance TangentSpace.module (x : M) : Module 𝕜 (TangentSpace I x) := by infer_instance
-
-/-- If two vector spaces are linearly equivalent, and one is finite-dimensional,
-    then so is the other. This version transfers the finite-dimensional property
-    via a continuous linear equivalence. -/
-lemma FiniteDimensional.of_equiv
-    {𝕜 : Type u} [RCLike 𝕜]
-    {E : Type v} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E]
-    {F : Type w} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F]
-    [FiniteDimensional 𝕜 E]
-    (e : E ≃L[𝕜] F) : FiniteDimensional 𝕜 F :=
-  LinearEquiv.finiteDimensional e.toLinearEquiv
-
-/-- If two vector spaces are linearly equivalent, and one is finite-dimensional,
-    then so is the other. This version works with a linear equivalence. -/
-lemma FiniteDimensional.of_linearEquiv
-    {𝕜 : Type u} [RCLike 𝕜]
-    {E : Type v} [AddCommGroup E] [Module 𝕜 E]
-    {F : Type w} [AddCommGroup F] [Module 𝕜 F]
-    [FiniteDimensional 𝕜 E]
-    (e : E ≃ₗ[𝕜] F) : FiniteDimensional 𝕜 F := by
-  exact Finite.equiv e
-
-lemma VectorBundle.finiteDimensional_fiber
-    (𝕜 : Type u) [RCLike 𝕜]
-    {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [FiniteDimensional 𝕜 F]
-    {B : Type w} [TopologicalSpace B]
-    {E_bundle : B → Type*} [∀ x, AddCommGroup (E_bundle x)] [∀ x, Module 𝕜 (E_bundle x)]
-    [TopologicalSpace (Bundle.TotalSpace F E_bundle)]
-    [∀ x, TopologicalSpace (E_bundle x)]
-    [FiberBundle F E_bundle] [VectorBundle 𝕜 F E_bundle]
-    (x : B) : FiniteDimensional 𝕜 (E_bundle x) :=
-  let triv := trivializationAt F E_bundle x
-  let hx := mem_baseSet_trivializationAt F E_bundle x
-  have h_linear : triv.IsLinear 𝕜 := VectorBundle.trivialization_linear' triv
-  haveI : triv.IsLinear 𝕜 := h_linear
-  let linear_equiv := triv.continuousLinearEquivAt 𝕜 x hx
-  FiniteDimensional.of_equiv linear_equiv.symm
-
-instance TangentSpace.finiteDimensional
-  [FiniteDimensional 𝕜 E]
-  [∀ x : M, TopologicalSpace (TangentSpace I x)]
-  [TopologicalSpace (Bundle.TotalSpace E (TangentSpace I : M → Type _))]
-  [FiberBundle E (TangentSpace I : M → Type _)]
-  [VectorBundle 𝕜 E (TangentSpace I : M → Type _)]
-  (x : M) :
-    FiniteDimensional 𝕜 (TangentSpace I x) :=
-  VectorBundle.finiteDimensional_fiber 𝕜 (F := E) (E_bundle := TangentSpace I) x
-
-variable
-    [inst_top : TopologicalSpace (TangentBundle I M)]
-    [inst_fib : FiberBundle E (TangentSpace I : M → Type _)]
-    [inst_vec : VectorBundle 𝕜 E (TangentSpace I : M → Type _)]
-    [inst_mani : IsManifold I (n + 1) M]
-    [inst_cmvb : ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
+variable {E : Type v} {H : Type w} {M : Type w} {n : WithTop ℕ∞}
+variable [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+variable [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E]
+variable {I : ModelWithCorners ℝ E H}
+variable [IsManifold I (n + 1) M]
+variable [TopologicalSpace (TotalSpace E (TangentSpace I : M → Type _))]
+variable [FiberBundle E (TangentSpace I : M → Type _)]
+variable [VectorBundle ℝ E (TangentSpace I : M → Type _)]
+variable [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
+variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
+variable {g : PseudoRiemannianMetric E H M n I}
 
-/-- Convert the continuous linear map representation `val x` to the algebraic `LinearMap.BilinForm`.
-    This is useful for leveraging lemmas about `BilinForm`. -/
-def toBilinForm
-    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
-         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
-    LinearMap.BilinForm 𝕜 (TangentSpace I x) :=
-  (g.val x).toBilinForm
-
-/-- Coercion from PseudoRiemannianMetric to its function representation. -/
-instance : CoeFun (@PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
-        inst_fib inst_vec inst_mani inst_cmvb)
-        (fun _ => ∀ x : M, TangentSpace I x →L[𝕜] (TangentSpace I x →L[𝕜] 𝕜)) where
-           coe g := g.val
-
-omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
-@[simp] lemma toBilinForm_apply
-    (g : PseudoRiemannianMetric I n) (x : M) (v w : TangentSpace I x) :
-  g.toBilinForm x v w = g.val x v w := rfl
-
-omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
-@[simp] lemma toBilinForm_isSymm
-    (g : PseudoRiemannianMetric I n) (x : M) : (g.toBilinForm x).IsSymm := by
-  intro v w
-  simp only [toBilinForm_apply]
-  exact g.symm x v w
-
-omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
-@[simp] lemma toBilinForm_nondegenerate
-    (g : PseudoRiemannianMetric I n) (x : M) :
-    (g.toBilinForm x).Nondegenerate := by
-  intro v hv
-  simp_rw [toBilinForm_apply] at hv
-  exact g.nondegenerate x v hv
-
-omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
-lemma symm'
-    (g : PseudoRiemannianMetric I n) (x : M) (v w : TangentSpace I x) :
-    (g.val x v) w = (g.val x w) v :=
+instance TangentSpace.addCommGroup (x : M) : AddCommGroup (TangentSpace I x) := by infer_instance
+instance TangentSpace.module (x : M) : Module ℝ (TangentSpace I x) := by infer_instance
+instance TangentSpace.finiteDimensional (x : M) : FiniteDimensional ℝ (TangentSpace I x) := by
+  infer_instance
+
+/-- Convert the metric's continuous linear map representation `val x` to the algebraic
+    `LinearMap.BilinForm`. -/
+def toBilinForm (g : PseudoRiemannianMetric E H M n I) (x : M) : LinearMap.BilinForm ℝ (TangentSpace I x) where
+  toFun := λ v => { toFun := λ w => g.val x v w,
+                    map_add' := λ w₁ w₂ => by
+                      simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.add_apply],
+                    map_smul' := λ c w => by
+                      simp only [map_smul, smul_eq_mul, RingHom.id_apply] }
+  map_add' := λ v₁ v₂ => by
+    ext w
+    simp only [map_add, add_apply, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
+  map_smul' := λ c v => by
+    ext w
+    simp only [map_smul, coe_smul', Pi.smul_apply, smul_eq_mul, LinearMap.coe_mk, AddHom.coe_mk,
+      RingHom.id_apply, LinearMap.smul_apply]
+
+/-- Convert a pseudo-Riemannian metric at a point `x` to a quadratic form `v ↦ gₓ(v, v)`. -/
+def toQuadraticForm (g : PseudoRiemannianMetric E H M n I) (x : M) : QuadraticForm ℝ (TangentSpace I x) :=
+  PseudoRiemannianMetricValToQuadraticForm g.val g.symm x
+
+-- Coercion from PseudoRiemannianMetric to its function representation.
+instance coeFunInst : CoeFun (PseudoRiemannianMetric E H M n I)
+        (fun _ => ∀ x : M, TangentSpace I x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ)) where
+   coe g := g.val
+
+@[simp] lemma toBilinForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
+  toBilinForm g x v w = g.val x v w := rfl
+
+@[simp] lemma toQuadraticForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) :
+  toQuadraticForm g x v = g.val x v v := rfl
+
+@[simp] lemma toBilinForm_isSymm (g : PseudoRiemannianMetric E H M n I) (x : M) : (toBilinForm g x).IsSymm := by
+  intro v w; simp only [toBilinForm_apply]; exact g.symm x v w
+
+@[simp] lemma toBilinForm_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x : M) : (toBilinForm g x).Nondegenerate := by
+  intro v hv; simp_rw [toBilinForm_apply] at hv; exact g.nondegenerate x v hv
+
+lemma symm' (x : M) (v w : TangentSpace I x) : (g.val x v) w = (g.val x w) v :=
   g.symm x v w
 
-omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
-lemma nondegenerate'
-    (g : PseudoRiemannianMetric I n) (x : M) (v : TangentSpace I x)
-    (h : ∀ w : TangentSpace I x, (g.val x v) w = 0) : v = 0 :=
+lemma nondegenerate' (x : M) (v : TangentSpace I x) (h : ∀ w : TangentSpace I x, (g.val x v) w = 0) :
+  v = 0 :=
   g.nondegenerate x v h
 
-omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
-lemma smooth'
-    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
-         inst_fib inst_vec inst_mani inst_cmvb) (X Y : ContMDiffSection I E n (TangentSpace I)) :
-    ContMDiff I (modelWithCornersSelf 𝕜 𝕜) n (fun x => (g.val x (X x)) (Y x)) :=
-  g.smooth X Y
-
-/-- Convert a pseudo-Riemannian metric at a point to a quadratic form. -/
-@[simp] def toQuadraticForm
-    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
-         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
-    QuadraticForm 𝕜 (TangentSpace I x) where
-  toFun v := g.val x v v
-  toFun_smul a v := by
-    simp only [ContinuousLinearMap.map_smul, ContinuousLinearMap.smul_apply, smul_smul, pow_two]
-  exists_companion' :=
-      ⟨ LinearMap.mk₂ 𝕜 (fun v y => g.val x v y + g.val x y v)
-        (fun v₁ v₂ y => by -- Additivity in v
-          simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.add_apply]
-          ring)
-        (fun a v y => by -- Homogeneity in v
-          simp only [ContinuousLinearMap.map_smul, ContinuousLinearMap.smul_apply]
-          ring_nf
-          exact Eq.symm (DistribSMul.smul_add a (((g.val x) v) y) (((g.val x) y) v)))
-        (fun v y₁ y₂ => by -- Additivity in y
-          simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.add_apply]
-          ring)
-        (fun a v y => by -- Homogeneity in y
-          simp only [ContinuousLinearMap.map_smul, ContinuousLinearMap.smul_apply]
-          ring_nf
-          exact Eq.symm (DistribSMul.smul_add a (((g.val x) v) y) (((g.val x) y) v))),
-            by
-              intro v y
-              simp only [LinearMap.mk₂_apply, ContinuousLinearMap.map_add,
-                ContinuousLinearMap.add_apply, g.symm x]
-              ring⟩
+lemma smooth' (x₀ : M) (v w : E) :
+  ContDiffWithinAt ℝ n (fun y => g.val ((extChartAt I x₀).symm y)
+    (mfderiv I I ((extChartAt I x₀).symm) y v) (mfderiv I I ((extChartAt I x₀).symm) y w))
+    ((extChartAt I x₀).target) ((extChartAt I x₀) x₀) :=
+  g.smooth_in_charts' x₀ v w
 
-omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
-lemma toQuadraticForm_apply
-    (g : PseudoRiemannianMetric I n) (x : M) (v : TangentSpace I x) :
-    g.toQuadraticForm x v = g.val x v v := rfl
+lemma negDim_isLocallyConstant' : IsLocallyConstant (fun x => (toQuadraticForm g x).negDim) :=
+  g.negDim_isLocallyConstant
 
 /-- The "musical" isomorphism (index lowering) from the tangent space to its dual,
     induced by a pseudo-Riemannian metric. -/
-def flat
-    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
-         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
-    TangentSpace I x →ₗ[𝕜] (TangentSpace I x →L[𝕜] 𝕜) :=
+def flat (g : PseudoRiemannianMetric E H M n I) (x : M) : TangentSpace I x →ₗ[ℝ] (TangentSpace I x →L[ℝ] ℝ) :=
   { toFun := λ v => g.val x v,
     map_add' := λ v w => by simp only [ContinuousLinearMap.map_add],
-    map_smul' := λ a v => by simp only [ContinuousLinearMap.map_smul]; exact rfl }
+    map_smul' := λ a v => by simp only [ContinuousLinearMap.map_smul]; rfl }
 
-omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
-@[simp] lemma flat_apply
-    (g : PseudoRiemannianMetric I n) (x : M) (v w : TangentSpace I x) :
-    (g.flat x v) w = g.val x v w := rfl
+@[simp] lemma flat_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
+  (flat g x v) w = g.val x v w := by rfl
 
 /-- The musical isomorphism as a continuous linear map. -/
-def flatL
-    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
-         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
-    TangentSpace I x →L[𝕜] (TangentSpace I x →L[𝕜] 𝕜) :=
-  { g.flat x with
+def flatL (g : PseudoRiemannianMetric E H M n I) (x : M) : TangentSpace I x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ) :=
+  { toFun := λ v => g.val x v,
+    map_add' := λ v w => by simp only [ContinuousLinearMap.map_add],
+    map_smul' := λ a v => by simp only [ContinuousLinearMap.map_smul]; rfl,
     cont := ContinuousLinearMap.continuous (g.val x) }
 
-omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
-@[simp] lemma flatL_apply
-    (g : PseudoRiemannianMetric I n) (x : M) (v w : TangentSpace I x) :
-    (g.flatL x v) w = g.val x v w := rfl
-
-omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
-/-- The linear map `flat` is injective due to non-degeneracy. -/
-@[simp] lemma flat_inj
-    (g : PseudoRiemannianMetric I n) (x : M) :
-    Function.Injective (g.flat x) := by
-  rw [← LinearMap.ker_eq_bot]
-  apply LinearMap.ker_eq_bot'.mpr
-  intro v hv
-  apply g.nondegenerate' x v
-  intro w
-  exact DFunLike.congr_fun hv w
-
-omit [FiniteDimensional 𝕜 E] [ChartedSpace H E] in
-/-- The continuous linear map `flatL` is injective. -/
-@[simp] lemma flatL_inj
-    (g : PseudoRiemannianMetric I n) (x : M) :
-    Function.Injective (g.flatL x) := by
-  intro v₁ v₂ h
-  apply flat_inj g x
-  exact h
-
-omit [ChartedSpace H E] in
-/-- The continuous linear map `flatL` is surjective because the tangent space is
-    finite dimensional. -/
+@[simp] lemma flatL_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
+  (flatL g x v) w = g.val x v w := rfl
+
+@[simp] lemma flat_inj (g : PseudoRiemannianMetric E H M n I) (x : M) : Function.Injective (flat g x) := by
+  rw [← LinearMap.ker_eq_bot]; apply LinearMap.ker_eq_bot'.mpr
+  intro v hv; apply g.nondegenerate' x v; intro w; exact DFunLike.congr_fun hv w
+
+@[simp] lemma flatL_inj (g : PseudoRiemannianMetric E H M n I) (x : M) : Function.Injective (flatL g x) :=
+  flat_inj g x -- Injective on LinearMap implies injective on ContLinearMap
+
+/-- In a finite-dimensional normed space, the continuous dual is linearly equivalent to the algebraic dual.
+    This is because in finite dimensions, every linear functional is continuous. -/
+def ContinuousLinearMap.equivModuleDual (𝕜 E : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
+    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] :
+    (E →L[𝕜] 𝕜) ≃ₗ[𝕜] Module.Dual 𝕜 E where
+  toFun f := f.toLinearMap
+  invFun φ :=
+    { toFun := φ
+      map_add' := LinearMap.map_add φ
+      map_smul' := LinearMap.map_smul φ
+      cont := φ.continuous_of_finiteDimensional }
+  map_add' f g := rfl
+  map_smul' c f := rfl
+  left_inv f := by ext; rfl
+  right_inv φ := rfl
+
+/-- Helper theorem: in finite dimensions, every linear map is continuous -/
+theorem continuous_linear_map_of_finite_dimension {𝕜 E : Type*} [NontriviallyNormedField 𝕜]
+    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] [CompleteSpace 𝕜]
+    (f : E →ₗ[𝕜] 𝕜) : Continuous f :=
+  by exact LinearMap.continuous_of_finiteDimensional f
+
+/-- For a finite-dimensional normed space, the dimension of the continuous dual
+equals the dimension of the original space. -/
+lemma finrank_continuousDual_eq_finrank {𝕜 E : Type*} [NontriviallyNormedField 𝕜]
+    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] [CompleteSpace 𝕜] :
+    finrank 𝕜 (E →L[𝕜] 𝕜) = finrank 𝕜 E := by
+  have h1 : (E →L[𝕜] 𝕜) ≃ₗ[𝕜] Module.Dual 𝕜 E := by
+    exact ContinuousLinearMap.equivModuleDual 𝕜 E
+  have h2 : finrank 𝕜 (Module.Dual 𝕜 E) = finrank 𝕜 E := by
+    exact finrank_linearMap_self 𝕜 𝕜 E
+  rw [LinearEquiv.finrank_eq h1, h2]
+
 @[simp] lemma flatL_surj
-    (g : PseudoRiemannianMetric I n) (x : M) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) :
     Function.Surjective (g.flatL x) := by
-  haveI : FiniteDimensional 𝕜 (TangentSpace I x) := TangentSpace.finiteDimensional x
-  have h_finrank_eq : finrank 𝕜 (TangentSpace I x) = finrank 𝕜 (TangentSpace I x →L[𝕜] 𝕜) := by
-    have h_dual_eq : finrank 𝕜 (TangentSpace I x →L[𝕜] 𝕜) = finrank 𝕜 (Module.Dual 𝕜
+  haveI : FiniteDimensional ℝ (TangentSpace I x) := TangentSpace.finiteDimensional x
+  have h_finrank_eq : finrank ℝ (TangentSpace I x) = finrank ℝ (TangentSpace I x →L[ℝ] ℝ) := by
+    have h_dual_eq : finrank ℝ (TangentSpace I x →L[ℝ] ℝ) = finrank ℝ (Module.Dual ℝ
     (TangentSpace I x)) := by
-      let to_dual : (TangentSpace I x →L[𝕜] 𝕜) → Module.Dual 𝕜 (TangentSpace I x) :=
+      let to_dual : (TangentSpace I x →L[ℝ] ℝ) → Module.Dual ℝ (TangentSpace I x) :=
         fun f => f.toLinearMap
-      let from_dual : Module.Dual 𝕜 (TangentSpace I x) → (TangentSpace I x →L[𝕜] 𝕜) := fun f =>
+      let from_dual : Module.Dual ℝ (TangentSpace I x) → (TangentSpace I x →L[ℝ] ℝ) := fun f =>
         ContinuousLinearMap.mk f (by
           apply LinearMap.continuous_of_finiteDimensional)
-      let equiv : (TangentSpace I x →L[𝕜] 𝕜) ≃ₗ[𝕜] Module.Dual 𝕜 (TangentSpace I x) :=
+      let equiv : (TangentSpace I x →L[ℝ] ℝ) ≃ₗ[ℝ] Module.Dual ℝ (TangentSpace I x) :=
       { toFun := to_dual,
         invFun := from_dual,
         map_add' := fun f g => by
@@ -332,103 +383,86 @@ omit [ChartedSpace H E] in
 /-- The "musical" isomorphism (index lowering) from the tangent space to its dual,
     as a continuous linear equivalence. -/
 @[simp] def flatEquiv
-    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
-         inst_fib inst_vec inst_mani inst_cmvb)
+    (g : PseudoRiemannianMetric E H M n I)
     (x : M) :
-    TangentSpace I x ≃L[𝕜] (TangentSpace I x →L[𝕜] 𝕜) :=
+    TangentSpace I x ≃L[ℝ] (TangentSpace I x →L[ℝ] ℝ) :=
   LinearEquiv.toContinuousLinearEquiv
     (LinearEquiv.ofBijective
       ((g.flatL x).toLinearMap)
       ⟨g.flatL_inj x, g.flatL_surj x⟩)
 
-omit [ChartedSpace H E] in
 lemma coe_flatEquiv
-    (g : PseudoRiemannianMetric I n) (x : M) :
-    (g.flatEquiv x : TangentSpace I x →ₗ[𝕜] (TangentSpace I x →L[𝕜] 𝕜)) = g.flatL x := rfl
+     (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (g.flatEquiv x : TangentSpace I x →ₗ[ℝ] (TangentSpace I x →L[ℝ] ℝ)) = g.flatL x := rfl
 
-omit [ChartedSpace H E] in
 @[simp] lemma flatEquiv_apply
-    (g : PseudoRiemannianMetric I n) (x : M) (v w : TangentSpace I x) :
+     (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
     (g.flatEquiv x v) w = g.val x v w := rfl
 
 /-- The "musical" isomorphism (index raising) from the dual of the tangent space to the
     tangent space, induced by a pseudo-Riemannian metric. This is the inverse of `flatEquiv`. -/
 def sharpEquiv
-    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
-         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
-    (TangentSpace I x →L[𝕜] 𝕜) ≃L[𝕜] TangentSpace I x :=
+     (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (TangentSpace I x →L[ℝ] ℝ) ≃L[ℝ] TangentSpace I x :=
   (g.flatEquiv x).symm
-#lint
+
 /-- The index raising map `sharp` as a continuous linear map. -/
 def sharpL
-    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
-         inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
-    (TangentSpace I x →L[𝕜] 𝕜) →L[𝕜] TangentSpace I x :=
+     (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (TangentSpace I x →L[ℝ] ℝ) →L[ℝ] TangentSpace I x :=
   (g.sharpEquiv x).toContinuousLinearMap
 
-omit [ChartedSpace H E] in
 lemma sharpL_eq_toContinuousLinearMap
-    (g : PseudoRiemannianMetric I n) (x : M) :
+     (g : PseudoRiemannianMetric E H M n I) (x : M) :
   g.sharpL x = (g.sharpEquiv x).toContinuousLinearMap := rfl
 
-omit [ChartedSpace H E] in
 lemma coe_sharpEquiv
-    (g : PseudoRiemannianMetric I n) (x : M) :
-    (g.sharpEquiv x : (TangentSpace I x →L[𝕜] 𝕜) →L[𝕜] TangentSpace I x) = g.sharpL x := rfl
+     (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (g.sharpEquiv x : (TangentSpace I x →L[ℝ] ℝ) →L[ℝ] TangentSpace I x) = g.sharpL x := rfl
 
 /-- The index raising map `sharp` as a linear map. -/
 noncomputable def sharp
-    (g : @PseudoRiemannianMetric 𝕜 _ E _ _ H _ M _ _ I n inst_top
-    inst_fib inst_vec inst_mani inst_cmvb) (x : M) :
-    (TangentSpace I x →L[𝕜] 𝕜) →ₗ[𝕜] TangentSpace I x :=
+     (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (TangentSpace I x →L[ℝ] ℝ) →ₗ[ℝ] TangentSpace I x :=
   (g.sharpEquiv x).toLinearEquiv.toLinearMap
 
-omit [ChartedSpace H E] in
 @[simp] lemma sharpL_apply_flatL
-    (g : PseudoRiemannianMetric I n) (x : M) (v : TangentSpace I x) :
+     (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) :
     g.sharpL x (g.flatL x v) = v :=
   (g.flatEquiv x).left_inv v
 
-omit [ChartedSpace H E] in
 @[simp] lemma flatL_apply_sharpL
-    (g : PseudoRiemannianMetric I n) (x : M) (ω : TangentSpace I x →L[𝕜] 𝕜) :
+     (g : PseudoRiemannianMetric E H M n I) (x : M) (ω : TangentSpace I x →L[ℝ] ℝ) :
     g.flatL x (g.sharpL x ω) = ω :=
   (g.flatEquiv x).right_inv ω
 
-omit [ChartedSpace H E] in
 /-- Applying `sharp` then `flat` recovers the original covector. -/
 @[simp] lemma flat_sharp_apply
-    (g : PseudoRiemannianMetric I n) (x : M) (ω : TangentSpace I x →L[𝕜] 𝕜) :
+     (g : PseudoRiemannianMetric E H M n I) (x : M) (ω : TangentSpace I x →L[ℝ] ℝ) :
     g.flat x (g.sharp x ω) = ω := by
-  -- Use the continuous versions and coerce
   have := flatL_apply_sharpL g x ω
-  -- Need to relate flat and flatL, sharp and sharpL
   simp only [sharp, sharpL, flat, flatL, coe_flatEquiv]; simp only [coe_sharpEquiv,
              ContinuousLinearEquiv.coe_coe, LinearEquiv.coe_coe] at this ⊢
   exact this
 
-omit [ChartedSpace H E] in
 @[simp] lemma sharp_flat_apply
-    (g : PseudoRiemannianMetric I n) (x : M) (v : TangentSpace I x) :
+     (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) :
     g.sharp x (g.flat x v) = v := by
-  -- Use the continuous versions and coerce
   have := sharpL_apply_flatL g x v
   simp only [sharp, sharpL, flat, flatL]; simp only [coe_flatEquiv, coe_sharpEquiv,
              ContinuousLinearEquiv.coe_coe, LinearEquiv.coe_coe] at this ⊢
   exact this
 
-omit [ChartedSpace H E] in
 /-- The metric evaluated at `sharp ω₁` and `sharp ω₂`. -/
 @[simp] lemma apply_sharp_sharp
-    (g : PseudoRiemannianMetric I n) (x : M) (ω₁ ω₂ : TangentSpace I x →L[𝕜] 𝕜) :
+     (g : PseudoRiemannianMetric E H M n I) (x : M) (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
     g.val x (g.sharpL x ω₁) (g.sharpL x ω₂) = ω₁ (g.sharpL x ω₂) := by
   rw [← flatL_apply g x (g.sharpL x ω₁)]
   rw [flatL_apply_sharpL g x ω₁]
 
-omit [ChartedSpace H E] in
 /-- The metric evaluated at `v` and `sharp ω`. -/
 lemma apply_vec_sharp
-    (g : PseudoRiemannianMetric I n) (x : M) (v : TangentSpace I x) (ω : TangentSpace I x →L[𝕜] 𝕜) :
+     (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) (ω : TangentSpace I x →L[ℝ] ℝ) :
     g.val x v (g.sharpL x ω) = ω v := by
   rw [g.symm' x v (g.sharpL x ω)]
   rw [← flatL_apply g x (g.sharpL x ω)]
diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index 21b8b5e86..38f930912 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -1,185 +1,105 @@
-/-
-Copyright (c) 2025 Matteo Cipollina. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Matteo Cipollina
--/
-
-import Mathlib.LinearAlgebra.QuadraticForm.Dual
 import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
+import Mathlib.LinearAlgebra.QuadraticForm.Dual
+import Mathlib.Analysis.InnerProductSpace.Basic
 
-/-!
-# Riemannian Metrics on Manifolds
-
-/-!
-This file introduces `RiemannianMetric`, a smooth positive‐definite specialization of
-`PseudoRiemannianMetric` over ℝ.  It requires that each bilinear form `gₓ` satisfy
-`gₓ(v,v) > 0` for nonzero tangent vectors `v`, and uses this to define the canonical
-inner product, norm, and distance on each tangent space.  (TODO: assemble these
-data into an `InnerProductSpace ℝ (TangentSpace I_ℝ x)` instance).
--/
-
-## Main Definitions
-
-*   `RiemannianMetric I n M`: A structure representing a `C^n` Riemannian metric on a manifold `M`
-    modelled on `(E_ℝ, H_ℝ)` with model `I_ℝ`. This specializes `PseudoRiemannianMetric` to the
-    case where the base field is `ℝ` and requires the bilinear form `gₓ` at each point `x` to be
-    positive-definite.
-*   `RiemannianMetric.inner g x`: The inner product (bilinear form `gₓ`) on the tangent space `TₓM`.
-*   `RiemannianMetric.sharp g x`: The musical isomorphism (index raising) from the cotangent space
-    `T*ₓM` to the tangent space `TₓM`, which is an isomorphism in the Riemannian case due to
-    positive definiteness.
-*   `RiemannianMetric.toQuadraticForm g x`: Expresses the metric at point `x` as a quadratic form.
-
-## Implementation Notes
-
-*   Extends `PseudoRiemannianMetric` by fixing the base field to `ℝ` and adding the `pos_def'`
-    field, ensuring `gₓ(v, v) > 0` for non-zero `v`.
-*   The positive definiteness allows defining an `InnerProductSpace ℝ (TₓM)` instance on each
-    tangent space `TₓM`.
--/
-universe u v w
+section RiemannianMetric
 
-open PseudoRiemannianMetric Bundle ContinuousLinearMap
+open Bundle Set Finset Function Filter Module Topology ContinuousLinearMap
+open scoped Manifold Bundle LinearMap Dual
+open PseudoRiemannianMetric InnerProductSpace
 
 noncomputable section
 
-variable {E_ℝ : Type v} [NormedAddCommGroup E_ℝ] [NormedSpace ℝ E_ℝ] [FiniteDimensional ℝ E_ℝ]
-variable {H_ℝ : Type w} [TopologicalSpace H_ℝ]
-variable {M_ℝ : Type w} [TopologicalSpace M_ℝ] [ChartedSpace H_ℝ M_ℝ] [ChartedSpace H_ℝ E_ℝ]
+variable {E : Type v} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+variable {H : Type w} [TopologicalSpace H]
+variable {M : Type w} [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E]
+variable {I : ModelWithCorners ℝ E H} {n : ℕ∞}
 
 /-- A Riemannian metric of smoothness class `n` on a manifold `M` over `ℝ`.
     This extends a pseudo-Riemannian metric with the positive definiteness condition. -/
 @[ext]
 structure RiemannianMetric
-  (I_ℝ : ModelWithCorners ℝ E_ℝ H_ℝ)
-  (n    : WithTop ℕ∞)
-  [inst_top : TopologicalSpace (TangentBundle I_ℝ M_ℝ)]
-  [inst_fib : FiberBundle      E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
-  [inst_vec : VectorBundle    ℝ E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
-  [inst_mani : IsManifold       I_ℝ (n + 1) M_ℝ]
-  [inst_cmvb : ContMDiffVectorBundle n E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _) I_ℝ]
-  extends @PseudoRiemannianMetric ℝ Real.instRCLike E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib
-    inst_vec inst_mani inst_cmvb where
+  (I : ModelWithCorners ℝ E H) (n : ℕ∞) (M : Type w)
+  [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (n + 1) M]
+  [inst_top : TopologicalSpace (TangentBundle I M)]
+  [inst_fib : FiberBundle E (TangentSpace I : M → Type _)]
+  [inst_vec : VectorBundle ℝ E (TangentSpace I : M → Type _)]
+  [inst_cmvb : ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
+  [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)] : Type _ where
+  /-- The underlying pseudo-Riemannian metric. -/
+  toPseudoRiemannianMetric : PseudoRiemannianMetric E H M (n) I
   /-- `gₓ(v, v) > 0` for all nonzero `v`. -/
-  pos_def' : ∀ x v, v ≠ 0 → val x v v > 0
-
+  pos_def' : ∀ x v, v ≠ 0 → toPseudoRiemannianMetric.val x v v > 0
 
 namespace RiemannianMetric
 
--- Propagate instance assumptions
-variable {I_ℝ : ModelWithCorners ℝ E_ℝ H_ℝ} {n : WithTop ℕ∞}
-variable [inst_top : TopologicalSpace (TangentBundle I_ℝ M_ℝ)]
-variable [inst_fib : FiberBundle E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
-variable [inst_vec : VectorBundle ℝ E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
-variable [inst_mani : IsManifold I_ℝ (n + 1) M_ℝ]
-variable [inst_cmvb : ContMDiffVectorBundle n E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _) I_ℝ]
+variable {I : ModelWithCorners ℝ E H} {n : ℕ∞} {M : Type w}
+variable [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (n + 1) M]
+variable [inst_top : TopologicalSpace (TangentBundle I M)]
+variable [inst_fib : FiberBundle E (TangentSpace I : M → Type _)]
+variable [inst_vec : VectorBundle ℝ E (TangentSpace I : M → Type _)]
+variable [inst_cmvb : ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
+variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
 
 /-- Coercion from RiemannianMetric to its underlying PseudoRiemannianMetric. -/
-instance : Coe (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
-      inst_cmvb)
-                (@PseudoRiemannianMetric ℝ Real.instRCLike E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top
-         inst_fib inst_vec inst_mani inst_cmvb) where
+instance : Coe (RiemannianMetric I n M) (PseudoRiemannianMetric E H M (n) I) where
   coe g := g.toPseudoRiemannianMetric
 
-omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
-@[simp] lemma pos_def (g : RiemannianMetric I_ℝ n) (x : M_ℝ) (v : TangentSpace I_ℝ x) (hv : v ≠ 0) :
+@[simp] lemma pos_def (g : RiemannianMetric I n M) (x : M) (v : TangentSpace I x)
+    (hv : v ≠ 0) :
   (g.toPseudoRiemannianMetric.val x v) v > 0 := g.pos_def' x v hv
 
 /-- The inverse of the musical isomorphism (index raising), which is an isomorphism
     in the Riemannian case. This is well-defined because the metric is positive definite. -/
 def sharp
-    (g : (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
-      inst_cmvb)) (x : M_ℝ) :
-    Module.Dual ℝ (TangentSpace I_ℝ x) ≃ₗ[ℝ] TangentSpace I_ℝ x :=
-  let bilin := g.toPseudoRiemannianMetric.toBilinForm x
-  have h_nondeg : bilin.Nondegenerate :=
-    g.toPseudoRiemannianMetric.toBilinForm_nondegenerate x
-  LinearEquiv.symm (bilin.toDual h_nondeg)
+    (g : RiemannianMetric I n M) (x : M) :
+    (TangentSpace I x →L[ℝ] ℝ) ≃L[ℝ] TangentSpace I x :=
+  g.toPseudoRiemannianMetric.sharpEquiv x
 
 /-- Express a Riemannian metric at a point as a quadratic form. -/
-abbrev toQuadraticForm
-    (g : (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
-      inst_cmvb))
-    (x : M_ℝ) :
-    QuadraticForm ℝ (TangentSpace I_ℝ x) :=
+abbrev toQuadraticForm (g : RiemannianMetric I n M) (x : M) :
+    QuadraticForm ℝ (TangentSpace I x) :=
   g.toPseudoRiemannianMetric.toQuadraticForm x
 
-omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
 /-- The quadratic form associated with a Riemannian metric is positive definite. -/
-lemma toQuadraticForm_posDef (g : RiemannianMetric I_ℝ n) (x : M_ℝ) :
+@[simp] lemma toQuadraticForm_posDef (g : RiemannianMetric I n M) (x : M) :
     (g.toQuadraticForm x).PosDef :=
   λ v hv => g.pos_def x v hv
 
-omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
 /-- The application of a Riemannian metric's quadratic form to a vector. -/
-lemma toQuadraticForm_apply (g : RiemannianMetric I_ℝ n) (x : M_ℝ)
-    (v : TangentSpace I_ℝ x) :
-    g.toQuadraticForm x v = g.val x v v := by
+@[simp] lemma toQuadraticForm_apply (g : RiemannianMetric I n M) (x : M)
+    (v : TangentSpace I x) :
+    g.toQuadraticForm x v = g.toPseudoRiemannianMetric.val x v v := by
   simp only [toQuadraticForm, PseudoRiemannianMetric.toQuadraticForm_apply]
 
-omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
-/-- A positive definite quadratic form is nondegenerate. -/
-lemma posDef_to_nondegenerate [Invertible (2 : ℝ)]
-    (g : RiemannianMetric I_ℝ n) (x : M_ℝ) :
-    (QuadraticMap.associated (R := ℝ) (N := ℝ) (g.toQuadraticForm x)).Nondegenerate := by
-  let Q := g.toQuadraticForm x
-  let B := QuadraticMap.associated (R := ℝ) (N := ℝ) Q
-  constructor
-  · intro v h_all_zero_w
-    specialize h_all_zero_w v
-    have h_assoc_self : B v v = Q v :=
-        QuadraticMap.associated_eq_self_apply ℝ Q v
-    rw [h_assoc_self] at h_all_zero_w
-    by_cases h_v_zero : v = 0
-    · exact h_v_zero
-    · have h_quad_pos : Q v > 0 := g.pos_def x v h_v_zero
-      linarith [h_all_zero_w, h_quad_pos]
-  · intro w h_all_zero_v
-    specialize h_all_zero_v w
-    have h_assoc_self : B w w = Q w :=
-        QuadraticMap.associated_eq_self_apply ℝ Q w
-    rw [h_assoc_self] at h_all_zero_v
-    by_cases h_w_zero : w = 0
-    · exact h_w_zero
-    · have h_quad_pos : Q w > 0 := g.pos_def x w h_w_zero
-      linarith [h_all_zero_v, h_quad_pos]
-
-/-- The isometry between `(Q.prod <| -Q)` and `QuadraticForm.dualProd`,
-    where `Q` is the quadratic form associated with the Riemannian metric at `x`. -/
-abbrev toDualProdIso [Invertible (2 : ℝ)] (g : RiemannianMetric I_ℝ n) (x : M_ℝ) :
-    ((g.toQuadraticForm x).prod <| -(g.toQuadraticForm x)) →qᵢ
-    (QuadraticForm.dualProd ℝ (TangentSpace I_ℝ x)) :=
-  QuadraticForm.toDualProd (g.toQuadraticForm x)
+theorem riemannian_metric_negDim_zero (g : RiemannianMetric I n M) (x : M) :
+    (g.toQuadraticForm x).negDim = 0 := by
+    apply QuadraticForm.rankNeg_eq_zero
+    exact g.toQuadraticForm_posDef x
 
 /-- The inner product on the tangent space at point `x` induced by the Riemannian metric `g`. -/
-def inner
-    (g : (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
-      inst_cmvb)) (x : M_ℝ) (v w : TangentSpace I_ℝ x) : ℝ :=
+def inner (g : RiemannianMetric I n M) (x : M) (v w : TangentSpace I x) : ℝ :=
   g.toPseudoRiemannianMetric.val x v w
 
-omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
-@[simp] lemma inner_apply (g : RiemannianMetric I_ℝ n) (x : M_ℝ) (v w : TangentSpace I_ℝ x) :
-  inner g x v w = g.val x v w := rfl
+@[simp] lemma inner_apply (g : RiemannianMetric I n M) (x : M) (v w : TangentSpace I x) :
+  inner g x v w = g.toPseudoRiemannianMetric.val x v w := rfl
 
-variable (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n _ _ _ _ _) (x : M_ℝ)
+variable (g : RiemannianMetric I n M) (x : M)
 
-omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
 /-- Pointwise symmetry of the inner product. -/
-lemma inner_symm (v w : TangentSpace I_ℝ x) :
+lemma inner_symm (v w : TangentSpace I x) :
     g.inner x v w = g.inner x w v := by
   simp only [inner_apply]
   exact g.toPseudoRiemannianMetric.symm' x v w
 
 /-- The inner product space core for the tangent space at a point, derived from the
 Riemannian metric. -/
-noncomputable def tangentInnerCore
-  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
-  (x : M_ℝ) :
-  InnerProductSpace.Core ℝ (TangentSpace I_ℝ x) where
+noncomputable def tangentInnerCore (g : RiemannianMetric I n M) (x : M) :
+    InnerProductSpace.Core ℝ (TangentSpace I x) where
   inner := λ v w => g.inner x v w
   conj_symm := λ v w => by
     simp only [inner_apply, conj_trivial]
-    exact g.toPseudoRiemannianMetric.symm' x w v
+    exact g.toPseudoRiemannianMetric.symm x w v
   nonneg_re := λ v => by
     simp only [inner_apply, RCLike.re_to_real]
     by_cases hv : v = 0
@@ -195,41 +115,64 @@ noncomputable def tangentInnerCore
     have h_pos : g.inner x v v > 0 := g.pos_def x v h_v_ne_zero
     linarith [h_inner_zero, h_pos]
 
-/-! ### Inner product space structure. -/
-
-/-- Normed additive commutative group structure on the tangent space at a point `x`,
-derived from the Riemannian metric `g`. -/
-noncomputable def TangentSpace.metricNormedAddCommGroup
-  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
-  (x : M_ℝ) :
-  NormedAddCommGroup (TangentSpace I_ℝ x) :=
-  @InnerProductSpace.Core.toNormedAddCommGroup ℝ (TangentSpace I_ℝ x) _ _ _ (tangentInnerCore g x)
-
-/-- The pre-inner product space core for the tangent space at a point,
-derived from the Riemannian metric. -/
-noncomputable def tangentPreInnerCore
-  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
-  (x : M_ℝ) :
-  PreInnerProductSpace.Core ℝ (TangentSpace I_ℝ x) where
-  inner := λ v w => g.inner x v w
-  conj_symm := λ v w => by
-    simp only [inner_apply, conj_trivial]
-    exact g.toPseudoRiemannianMetric.symm' x w v
-  nonneg_re := λ v => by
-    simp only [inner_apply, RCLike.re_to_real]
-    by_cases hv : v = 0
-    · simp [hv, inner_apply, map_zero, zero_apply]
-    · exact le_of_lt (g.pos_def x v hv)
-  add_left := λ u v w => by
-    simp only [inner_apply, map_add, add_apply]
-  smul_left := λ r u v => by
-    simp only [inner_apply, map_smul, smul_apply, conj_trivial]
-    rfl
-
-/-- Each tangent space carries a seminormed add comm group structure derived from
-the Riemannian metric -/
-noncomputable def TangentSpace.metricSeminormedAddCommGroup
-  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
-  (x : M_ℝ) :
-  SeminormedAddCommGroup (TangentSpace I_ℝ x) :=
-  (TangentSpace.metricNormedAddCommGroup g x).toSeminormedAddCommGroup
+/- We Define NormedAddCommGroup and InnerProductSpace structures locally.
+ We DO NOT mark them as instance.-/
+
+/-- Creates a `NormedAddCommGroup` structure on the tangent space at a point,
+    derived from a Riemannian metric. -/
+noncomputable def TangentSpace.metricNormedAddCommGroup (g : RiemannianMetric I n M) (x : M) :
+    NormedAddCommGroup (TangentSpace I x) :=
+  @InnerProductSpace.Core.toNormedAddCommGroup ℝ (TangentSpace I x) _ _ _ (tangentInnerCore g x)
+
+/-- Creates an `InnerProductSpace` structure on the tangent space at a point,
+    derived from a Riemannian metric. Alternative implementation using `letI`. -/
+noncomputable def TangentSpace.metricInnerProductSpace' (g : RiemannianMetric I n M) (x : M) :
+    letI := TangentSpace.metricNormedAddCommGroup g x
+    InnerProductSpace ℝ (TangentSpace I x) :=
+  InnerProductSpace.ofCore (tangentInnerCore g x)
+
+/-- Creates an `InnerProductSpace` structure on the tangent space at a point,
+    derived from a Riemannian metric. -/
+noncomputable def TangentSpace.metricInnerProductSpace (g : RiemannianMetric I n M) (x : M) :
+    let _ := TangentSpace.metricNormedAddCommGroup g x
+    InnerProductSpace ℝ (TangentSpace I x) :=
+  let inner_core := tangentInnerCore g x
+  let _ := TangentSpace.metricNormedAddCommGroup g x
+  InnerProductSpace.ofCore inner_core
+
+/-- The norm on a tangent space induced by a Riemannian metric, defined as the square root
+    of the inner product of a vector with itself. -/
+noncomputable def norm (g : RiemannianMetric I n M) (x : M) (v : TangentSpace I x) : ℝ :=
+  Real.sqrt (g.inner x v v)
+
+-- Example using the norm function
+example (g : RiemannianMetric I n M) (x : M) (v : TangentSpace I x) :
+    norm g x v ≥ 0 := Real.sqrt_nonneg _
+
+-- Example showing how to use the metric inner product space
+example (g : RiemannianMetric I n M) (x : M) (v w : TangentSpace I x) :
+    (TangentSpace.metricInnerProductSpace g x).inner v w = g.inner x v w := by
+  letI := TangentSpace.metricInnerProductSpace g x
+  rfl
+
+/-- Helper function to compute the norm on a tangent space from a Riemannian metric,
+    using the underlying `NormedAddCommGroup` structure. -/
+noncomputable def norm' (g : RiemannianMetric I n M) (x : M) (v : TangentSpace I x) : ℝ :=
+  let normed_group := TangentSpace.metricNormedAddCommGroup g x
+  @Norm.norm (TangentSpace I x) (@NormedAddCommGroup.toNorm (TangentSpace I x) normed_group) v
+
+-- Example: Using a custom norm function instead of the notation
+example (g : RiemannianMetric I n M) (x : M) (v : TangentSpace I x) :
+    norm g x v ≥ 0 := by
+  unfold norm
+  apply Real.sqrt_nonneg
+
+example (g : RiemannianMetric I n M) (x : M) (v : TangentSpace I x) : ℝ :=
+  letI := TangentSpace.metricNormedAddCommGroup g x
+  ‖v‖
+
+example (g : RiemannianMetric I n M) (x : M) (v : TangentSpace I x) : ℝ :=
+  let normed_group := TangentSpace.metricNormedAddCommGroup g x
+  @Norm.norm (TangentSpace I x) (@NormedAddCommGroup.toNorm (TangentSpace I x) normed_group) v
+
+end RiemannianMetric
diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/DefsOrig.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/DefsOrig.lean
new file mode 100644
index 000000000..21b8b5e86
--- /dev/null
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/DefsOrig.lean
@@ -0,0 +1,235 @@
+/-
+Copyright (c) 2025 Matteo Cipollina. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Matteo Cipollina
+-/
+
+import Mathlib.LinearAlgebra.QuadraticForm.Dual
+import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
+
+/-!
+# Riemannian Metrics on Manifolds
+
+/-!
+This file introduces `RiemannianMetric`, a smooth positive‐definite specialization of
+`PseudoRiemannianMetric` over ℝ.  It requires that each bilinear form `gₓ` satisfy
+`gₓ(v,v) > 0` for nonzero tangent vectors `v`, and uses this to define the canonical
+inner product, norm, and distance on each tangent space.  (TODO: assemble these
+data into an `InnerProductSpace ℝ (TangentSpace I_ℝ x)` instance).
+-/
+
+## Main Definitions
+
+*   `RiemannianMetric I n M`: A structure representing a `C^n` Riemannian metric on a manifold `M`
+    modelled on `(E_ℝ, H_ℝ)` with model `I_ℝ`. This specializes `PseudoRiemannianMetric` to the
+    case where the base field is `ℝ` and requires the bilinear form `gₓ` at each point `x` to be
+    positive-definite.
+*   `RiemannianMetric.inner g x`: The inner product (bilinear form `gₓ`) on the tangent space `TₓM`.
+*   `RiemannianMetric.sharp g x`: The musical isomorphism (index raising) from the cotangent space
+    `T*ₓM` to the tangent space `TₓM`, which is an isomorphism in the Riemannian case due to
+    positive definiteness.
+*   `RiemannianMetric.toQuadraticForm g x`: Expresses the metric at point `x` as a quadratic form.
+
+## Implementation Notes
+
+*   Extends `PseudoRiemannianMetric` by fixing the base field to `ℝ` and adding the `pos_def'`
+    field, ensuring `gₓ(v, v) > 0` for non-zero `v`.
+*   The positive definiteness allows defining an `InnerProductSpace ℝ (TₓM)` instance on each
+    tangent space `TₓM`.
+-/
+universe u v w
+
+open PseudoRiemannianMetric Bundle ContinuousLinearMap
+
+noncomputable section
+
+variable {E_ℝ : Type v} [NormedAddCommGroup E_ℝ] [NormedSpace ℝ E_ℝ] [FiniteDimensional ℝ E_ℝ]
+variable {H_ℝ : Type w} [TopologicalSpace H_ℝ]
+variable {M_ℝ : Type w} [TopologicalSpace M_ℝ] [ChartedSpace H_ℝ M_ℝ] [ChartedSpace H_ℝ E_ℝ]
+
+/-- A Riemannian metric of smoothness class `n` on a manifold `M` over `ℝ`.
+    This extends a pseudo-Riemannian metric with the positive definiteness condition. -/
+@[ext]
+structure RiemannianMetric
+  (I_ℝ : ModelWithCorners ℝ E_ℝ H_ℝ)
+  (n    : WithTop ℕ∞)
+  [inst_top : TopologicalSpace (TangentBundle I_ℝ M_ℝ)]
+  [inst_fib : FiberBundle      E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
+  [inst_vec : VectorBundle    ℝ E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
+  [inst_mani : IsManifold       I_ℝ (n + 1) M_ℝ]
+  [inst_cmvb : ContMDiffVectorBundle n E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _) I_ℝ]
+  extends @PseudoRiemannianMetric ℝ Real.instRCLike E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib
+    inst_vec inst_mani inst_cmvb where
+  /-- `gₓ(v, v) > 0` for all nonzero `v`. -/
+  pos_def' : ∀ x v, v ≠ 0 → val x v v > 0
+
+
+namespace RiemannianMetric
+
+-- Propagate instance assumptions
+variable {I_ℝ : ModelWithCorners ℝ E_ℝ H_ℝ} {n : WithTop ℕ∞}
+variable [inst_top : TopologicalSpace (TangentBundle I_ℝ M_ℝ)]
+variable [inst_fib : FiberBundle E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
+variable [inst_vec : VectorBundle ℝ E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
+variable [inst_mani : IsManifold I_ℝ (n + 1) M_ℝ]
+variable [inst_cmvb : ContMDiffVectorBundle n E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _) I_ℝ]
+
+/-- Coercion from RiemannianMetric to its underlying PseudoRiemannianMetric. -/
+instance : Coe (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
+      inst_cmvb)
+                (@PseudoRiemannianMetric ℝ Real.instRCLike E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top
+         inst_fib inst_vec inst_mani inst_cmvb) where
+  coe g := g.toPseudoRiemannianMetric
+
+omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
+@[simp] lemma pos_def (g : RiemannianMetric I_ℝ n) (x : M_ℝ) (v : TangentSpace I_ℝ x) (hv : v ≠ 0) :
+  (g.toPseudoRiemannianMetric.val x v) v > 0 := g.pos_def' x v hv
+
+/-- The inverse of the musical isomorphism (index raising), which is an isomorphism
+    in the Riemannian case. This is well-defined because the metric is positive definite. -/
+def sharp
+    (g : (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
+      inst_cmvb)) (x : M_ℝ) :
+    Module.Dual ℝ (TangentSpace I_ℝ x) ≃ₗ[ℝ] TangentSpace I_ℝ x :=
+  let bilin := g.toPseudoRiemannianMetric.toBilinForm x
+  have h_nondeg : bilin.Nondegenerate :=
+    g.toPseudoRiemannianMetric.toBilinForm_nondegenerate x
+  LinearEquiv.symm (bilin.toDual h_nondeg)
+
+/-- Express a Riemannian metric at a point as a quadratic form. -/
+abbrev toQuadraticForm
+    (g : (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
+      inst_cmvb))
+    (x : M_ℝ) :
+    QuadraticForm ℝ (TangentSpace I_ℝ x) :=
+  g.toPseudoRiemannianMetric.toQuadraticForm x
+
+omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
+/-- The quadratic form associated with a Riemannian metric is positive definite. -/
+lemma toQuadraticForm_posDef (g : RiemannianMetric I_ℝ n) (x : M_ℝ) :
+    (g.toQuadraticForm x).PosDef :=
+  λ v hv => g.pos_def x v hv
+
+omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
+/-- The application of a Riemannian metric's quadratic form to a vector. -/
+lemma toQuadraticForm_apply (g : RiemannianMetric I_ℝ n) (x : M_ℝ)
+    (v : TangentSpace I_ℝ x) :
+    g.toQuadraticForm x v = g.val x v v := by
+  simp only [toQuadraticForm, PseudoRiemannianMetric.toQuadraticForm_apply]
+
+omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
+/-- A positive definite quadratic form is nondegenerate. -/
+lemma posDef_to_nondegenerate [Invertible (2 : ℝ)]
+    (g : RiemannianMetric I_ℝ n) (x : M_ℝ) :
+    (QuadraticMap.associated (R := ℝ) (N := ℝ) (g.toQuadraticForm x)).Nondegenerate := by
+  let Q := g.toQuadraticForm x
+  let B := QuadraticMap.associated (R := ℝ) (N := ℝ) Q
+  constructor
+  · intro v h_all_zero_w
+    specialize h_all_zero_w v
+    have h_assoc_self : B v v = Q v :=
+        QuadraticMap.associated_eq_self_apply ℝ Q v
+    rw [h_assoc_self] at h_all_zero_w
+    by_cases h_v_zero : v = 0
+    · exact h_v_zero
+    · have h_quad_pos : Q v > 0 := g.pos_def x v h_v_zero
+      linarith [h_all_zero_w, h_quad_pos]
+  · intro w h_all_zero_v
+    specialize h_all_zero_v w
+    have h_assoc_self : B w w = Q w :=
+        QuadraticMap.associated_eq_self_apply ℝ Q w
+    rw [h_assoc_self] at h_all_zero_v
+    by_cases h_w_zero : w = 0
+    · exact h_w_zero
+    · have h_quad_pos : Q w > 0 := g.pos_def x w h_w_zero
+      linarith [h_all_zero_v, h_quad_pos]
+
+/-- The isometry between `(Q.prod <| -Q)` and `QuadraticForm.dualProd`,
+    where `Q` is the quadratic form associated with the Riemannian metric at `x`. -/
+abbrev toDualProdIso [Invertible (2 : ℝ)] (g : RiemannianMetric I_ℝ n) (x : M_ℝ) :
+    ((g.toQuadraticForm x).prod <| -(g.toQuadraticForm x)) →qᵢ
+    (QuadraticForm.dualProd ℝ (TangentSpace I_ℝ x)) :=
+  QuadraticForm.toDualProd (g.toQuadraticForm x)
+
+/-- The inner product on the tangent space at point `x` induced by the Riemannian metric `g`. -/
+def inner
+    (g : (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
+      inst_cmvb)) (x : M_ℝ) (v w : TangentSpace I_ℝ x) : ℝ :=
+  g.toPseudoRiemannianMetric.val x v w
+
+omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
+@[simp] lemma inner_apply (g : RiemannianMetric I_ℝ n) (x : M_ℝ) (v w : TangentSpace I_ℝ x) :
+  inner g x v w = g.val x v w := rfl
+
+variable (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n _ _ _ _ _) (x : M_ℝ)
+
+omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
+/-- Pointwise symmetry of the inner product. -/
+lemma inner_symm (v w : TangentSpace I_ℝ x) :
+    g.inner x v w = g.inner x w v := by
+  simp only [inner_apply]
+  exact g.toPseudoRiemannianMetric.symm' x v w
+
+/-- The inner product space core for the tangent space at a point, derived from the
+Riemannian metric. -/
+noncomputable def tangentInnerCore
+  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
+  (x : M_ℝ) :
+  InnerProductSpace.Core ℝ (TangentSpace I_ℝ x) where
+  inner := λ v w => g.inner x v w
+  conj_symm := λ v w => by
+    simp only [inner_apply, conj_trivial]
+    exact g.toPseudoRiemannianMetric.symm' x w v
+  nonneg_re := λ v => by
+    simp only [inner_apply, RCLike.re_to_real]
+    by_cases hv : v = 0
+    · simp [hv, inner_apply, map_zero, zero_apply]
+    · exact le_of_lt (g.pos_def x v hv)
+  add_left := λ u v w => by
+    simp only [inner_apply, map_add, add_apply]
+  smul_left := λ r u v => by
+    simp only [inner_apply, map_smul, smul_apply, conj_trivial]
+    rfl
+  definite := fun v (h_inner_zero : g.inner x v v = 0) => by
+    by_contra h_v_ne_zero
+    have h_pos : g.inner x v v > 0 := g.pos_def x v h_v_ne_zero
+    linarith [h_inner_zero, h_pos]
+
+/-! ### Inner product space structure. -/
+
+/-- Normed additive commutative group structure on the tangent space at a point `x`,
+derived from the Riemannian metric `g`. -/
+noncomputable def TangentSpace.metricNormedAddCommGroup
+  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
+  (x : M_ℝ) :
+  NormedAddCommGroup (TangentSpace I_ℝ x) :=
+  @InnerProductSpace.Core.toNormedAddCommGroup ℝ (TangentSpace I_ℝ x) _ _ _ (tangentInnerCore g x)
+
+/-- The pre-inner product space core for the tangent space at a point,
+derived from the Riemannian metric. -/
+noncomputable def tangentPreInnerCore
+  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
+  (x : M_ℝ) :
+  PreInnerProductSpace.Core ℝ (TangentSpace I_ℝ x) where
+  inner := λ v w => g.inner x v w
+  conj_symm := λ v w => by
+    simp only [inner_apply, conj_trivial]
+    exact g.toPseudoRiemannianMetric.symm' x w v
+  nonneg_re := λ v => by
+    simp only [inner_apply, RCLike.re_to_real]
+    by_cases hv : v = 0
+    · simp [hv, inner_apply, map_zero, zero_apply]
+    · exact le_of_lt (g.pos_def x v hv)
+  add_left := λ u v w => by
+    simp only [inner_apply, map_add, add_apply]
+  smul_left := λ r u v => by
+    simp only [inner_apply, map_smul, smul_apply, conj_trivial]
+    rfl
+
+/-- Each tangent space carries a seminormed add comm group structure derived from
+the Riemannian metric -/
+noncomputable def TangentSpace.metricSeminormedAddCommGroup
+  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
+  (x : M_ℝ) :
+  SeminormedAddCommGroup (TangentSpace I_ℝ x) :=
+  (TangentSpace.metricNormedAddCommGroup g x).toSeminormedAddCommGroup

From 422e7bd7204eafe54c12bf0462e390a00904df74 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Tue, 29 Apr 2025 01:43:16 +0200
Subject: [PATCH 10/32] Update Defs.lean

---
 .../Metric/PseudoRiemannian/Defs.lean         | 51 ++++++++++++-------
 1 file changed, 33 insertions(+), 18 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index 9ecbb1b6c..22d87d9f9 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -73,11 +73,13 @@ lemma Pi.basisFun_apply_same {𝕜 ι : Type*} [Field 𝕜] [Fintype ι] [Decida
   simp only [Pi.basisFun_apply, Pi.single_eq_same]
 
 /-- The i-th standard basis vector has 0 in all positions j ≠ i. -/
-lemma Pi.basisFun_apply_ne {𝕜 ι : Type*} [Field 𝕜] [Fintype ι] [DecidableEq ι] (i j : ι) (h : j ≠ i) :
+lemma Pi.basisFun_apply_ne {𝕜 ι : Type*} [Field 𝕜] [Fintype ι] [DecidableEq ι] (i j : ι)
+  (h : j ≠ i) :
   (Pi.basisFun 𝕜 ι) i j = 0 := by
   simp only [Pi.basisFun_apply, Pi.single_eq_of_ne h]
 
-/-- For a standard basis vector in a weighted sum of squares, only one term in the sum is nonzero. -/
+/-- For a standard basis vector in a weighted sum of squares, only one term in the sum
+    is nonzero. -/
 lemma QuadraticMap.weightedSumSquares_basis_vector {E : Type*} [AddCommGroup E]
     [Module ℝ E] {weights : Fin (finrank ℝ E) → ℝ}
     {i : Fin (finrank ℝ E)} (v : Fin (finrank ℝ E) → ℝ)
@@ -143,7 +145,8 @@ theorem rankNeg_eq_zero {E : Type*} [AddCommGroup E]
   unfold QuadraticForm.negDim
   have h_exists := equivalent_signType_weighted_sum_squared q
   let w := Classical.choose h_exists
-  have h_equiv : QuadraticMap.Equivalent q (QuadraticMap.weightedSumSquares ℝ fun i => (w i : ℝ)) :=
+  have h_equiv : QuadraticMap.Equivalent q
+      (QuadraticMap.weightedSumSquares ℝ fun i => (w i : ℝ)) :=
     Classical.choose_spec h_exists
   have h_no_neg : ∀ i, w i ≠ SignType.neg :=
     QuadraticForm.posDef_no_neg_weights hq h_equiv
@@ -242,7 +245,8 @@ instance TangentSpace.finiteDimensional (x : M) : FiniteDimensional ℝ (Tangent
 
 /-- Convert the metric's continuous linear map representation `val x` to the algebraic
     `LinearMap.BilinForm`. -/
-def toBilinForm (g : PseudoRiemannianMetric E H M n I) (x : M) : LinearMap.BilinForm ℝ (TangentSpace I x) where
+def toBilinForm (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    LinearMap.BilinForm ℝ (TangentSpace I x) where
   toFun := λ v => { toFun := λ w => g.val x v w,
                     map_add' := λ w₁ w₂ => by
                       simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.add_apply],
@@ -257,7 +261,8 @@ def toBilinForm (g : PseudoRiemannianMetric E H M n I) (x : M) : LinearMap.Bilin
       RingHom.id_apply, LinearMap.smul_apply]
 
 /-- Convert a pseudo-Riemannian metric at a point `x` to a quadratic form `v ↦ gₓ(v, v)`. -/
-def toQuadraticForm (g : PseudoRiemannianMetric E H M n I) (x : M) : QuadraticForm ℝ (TangentSpace I x) :=
+def toQuadraticForm (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    QuadraticForm ℝ (TangentSpace I x) :=
   PseudoRiemannianMetricValToQuadraticForm g.val g.symm x
 
 -- Coercion from PseudoRiemannianMetric to its function representation.
@@ -265,22 +270,27 @@ instance coeFunInst : CoeFun (PseudoRiemannianMetric E H M n I)
         (fun _ => ∀ x : M, TangentSpace I x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ)) where
    coe g := g.val
 
-@[simp] lemma toBilinForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
+@[simp] lemma toBilinForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M)
+    (v w : TangentSpace I x) :
   toBilinForm g x v w = g.val x v w := rfl
 
-@[simp] lemma toQuadraticForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) :
+@[simp] lemma toQuadraticForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M)
+    (v : TangentSpace I x) :
   toQuadraticForm g x v = g.val x v v := rfl
 
-@[simp] lemma toBilinForm_isSymm (g : PseudoRiemannianMetric E H M n I) (x : M) : (toBilinForm g x).IsSymm := by
+@[simp] lemma toBilinForm_isSymm (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (toBilinForm g x).IsSymm := by
   intro v w; simp only [toBilinForm_apply]; exact g.symm x v w
 
-@[simp] lemma toBilinForm_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x : M) : (toBilinForm g x).Nondegenerate := by
+@[simp] lemma toBilinForm_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (toBilinForm g x).Nondegenerate := by
   intro v hv; simp_rw [toBilinForm_apply] at hv; exact g.nondegenerate x v hv
 
 lemma symm' (x : M) (v w : TangentSpace I x) : (g.val x v) w = (g.val x w) v :=
   g.symm x v w
 
-lemma nondegenerate' (x : M) (v : TangentSpace I x) (h : ∀ w : TangentSpace I x, (g.val x v) w = 0) :
+lemma nondegenerate' (x : M) (v : TangentSpace I x)
+   (h : ∀ w : TangentSpace I x, (g.val x v) w = 0) :
   v = 0 :=
   g.nondegenerate x v h
 
@@ -295,7 +305,8 @@ lemma negDim_isLocallyConstant' : IsLocallyConstant (fun x => (toQuadraticForm g
 
 /-- The "musical" isomorphism (index lowering) from the tangent space to its dual,
     induced by a pseudo-Riemannian metric. -/
-def flat (g : PseudoRiemannianMetric E H M n I) (x : M) : TangentSpace I x →ₗ[ℝ] (TangentSpace I x →L[ℝ] ℝ) :=
+def flat (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    TangentSpace I x →ₗ[ℝ] (TangentSpace I x →L[ℝ] ℝ) :=
   { toFun := λ v => g.val x v,
     map_add' := λ v w => by simp only [ContinuousLinearMap.map_add],
     map_smul' := λ a v => by simp only [ContinuousLinearMap.map_smul]; rfl }
@@ -304,7 +315,8 @@ def flat (g : PseudoRiemannianMetric E H M n I) (x : M) : TangentSpace I x →
   (flat g x v) w = g.val x v w := by rfl
 
 /-- The musical isomorphism as a continuous linear map. -/
-def flatL (g : PseudoRiemannianMetric E H M n I) (x : M) : TangentSpace I x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ) :=
+def flatL (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    TangentSpace I x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ) :=
   { toFun := λ v => g.val x v,
     map_add' := λ v w => by simp only [ContinuousLinearMap.map_add],
     map_smul' := λ a v => by simp only [ContinuousLinearMap.map_smul]; rfl,
@@ -313,15 +325,17 @@ def flatL (g : PseudoRiemannianMetric E H M n I) (x : M) : TangentSpace I x →L
 @[simp] lemma flatL_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
   (flatL g x v) w = g.val x v w := rfl
 
-@[simp] lemma flat_inj (g : PseudoRiemannianMetric E H M n I) (x : M) : Function.Injective (flat g x) := by
+@[simp] lemma flat_inj (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    Function.Injective (flat g x) := by
   rw [← LinearMap.ker_eq_bot]; apply LinearMap.ker_eq_bot'.mpr
   intro v hv; apply g.nondegenerate' x v; intro w; exact DFunLike.congr_fun hv w
 
-@[simp] lemma flatL_inj (g : PseudoRiemannianMetric E H M n I) (x : M) : Function.Injective (flatL g x) :=
-  flat_inj g x -- Injective on LinearMap implies injective on ContLinearMap
+@[simp] lemma flatL_inj (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    Function.Injective (flatL g x) :=
+  flat_inj g x
 
-/-- In a finite-dimensional normed space, the continuous dual is linearly equivalent to the algebraic dual.
-    This is because in finite dimensions, every linear functional is continuous. -/
+/-- In a finite-dimensional normed space, the continuous dual is linearly equivalent
+    to the algebraic dual. -/
 def ContinuousLinearMap.equivModuleDual (𝕜 E : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
     [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] :
     (E →L[𝕜] 𝕜) ≃ₗ[𝕜] Module.Dual 𝕜 E where
@@ -462,7 +476,8 @@ noncomputable def sharp
 
 /-- The metric evaluated at `v` and `sharp ω`. -/
 lemma apply_vec_sharp
-     (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) (ω : TangentSpace I x →L[ℝ] ℝ) :
+     (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x)
+     (ω : TangentSpace I x →L[ℝ] ℝ) :
     g.val x v (g.sharpL x ω) = ω v := by
   rw [g.symm' x v (g.sharpL x ω)]
   rw [← flatL_apply g x (g.sharpL x ω)]

From caf4d53e75b3fac17a7c2799e6575029188e2ee9 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Tue, 29 Apr 2025 01:48:27 +0200
Subject: [PATCH 11/32] fixes

---
 .../Geometry/Metric/Riemannian/Defs.lean      |  2 -
 .../Geometry/Metric/Riemannian/DefsOrig.lean  | 40 ++++++++++++++++++-
 2 files changed, 38 insertions(+), 4 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index 38f930912..30bdd2405 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -1,7 +1,5 @@
 import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
 import Mathlib.LinearAlgebra.QuadraticForm.Dual
-import Mathlib.Analysis.InnerProductSpace.Basic
-
 section RiemannianMetric
 
 open Bundle Set Finset Function Filter Module Topology ContinuousLinearMap
diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/DefsOrig.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/DefsOrig.lean
index 21b8b5e86..410c850cd 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/DefsOrig.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/DefsOrig.lean
@@ -63,10 +63,8 @@ structure RiemannianMetric
   /-- `gₓ(v, v) > 0` for all nonzero `v`. -/
   pos_def' : ∀ x v, v ≠ 0 → val x v v > 0
 
-
 namespace RiemannianMetric
 
--- Propagate instance assumptions
 variable {I_ℝ : ModelWithCorners ℝ E_ℝ H_ℝ} {n : WithTop ℕ∞}
 variable [inst_top : TopologicalSpace (TangentBundle I_ℝ M_ℝ)]
 variable [inst_fib : FiberBundle E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
@@ -233,3 +231,41 @@ noncomputable def TangentSpace.metricSeminormedAddCommGroup
   (x : M_ℝ) :
   SeminormedAddCommGroup (TangentSpace I_ℝ x) :=
   (TangentSpace.metricNormedAddCommGroup g x).toSeminormedAddCommGroup
+/-
+noncomputable def TangentSpace.metricInnerProductSpace
+  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
+  (x : M_ℝ) :
+  InnerProductSpace ℝ (TangentSpace I_ℝ x) := by
+
+  -- Ensure the underlying group structure is known
+  letI : AddCommGroup (TangentSpace I_ℝ x) := TangentSpace.addCommGroup x
+  letI : TopologicalSpace (TangentSpace I_ℝ x) := inferInstance
+  letI : TopologicalAddGroup (TangentSpace I_ℝ x) := inferInstance
+  letI : UniformSpace (TangentSpace I_ℝ x) := inferInstance
+  letI : UniformAddGroup (TangentSpace I_ℝ x) := ⟨⟩
+
+  -- Provide the specific SeminormedAddCommGroup instance
+  letI : SeminormedAddCommGroup (TangentSpace I_ℝ x) :=
+    TangentSpace.metricSeminormedAddCommGroup g x
+
+  -- Provide the NormedAddCommGroup instance
+  letI : NormedAddCommGroup (TangentSpace I_ℝ x) :=
+    TangentSpace.metricNormedAddCommGroup g x
+
+  -- Build the InnerProductSpace using the custom inner core
+  let core := tangentInnerCore g x
+  letI : NormedSpace ℝ (TangentSpace I_ℝ x) :=
+    @InnerProductSpace.Core.toNormedSpace ℝ (TangentSpace I_ℝ x) _ _ _ core
+  exact InnerProductSpace.ofCore core
+
+
+-- Check the norm_sq_eq_inner property holds for the constructed structure
+lemma TangentSpace.metric_norm_sq_eq_inner
+  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
+  (x : M_ℝ) (v : TangentSpace I_ℝ x) :
+  letI := TangentSpace.metricNormedAddCommGroup g x
+  ‖v‖ ^ 2 = g.inner x v v := by
+  letI := TangentSpace.metricInnerProductSpace g x
+  exact InnerProductSpace.norm_sq_eq_inner v
+-/
+end RiemannianMetric

From b158d58f60694318df0661e9bac1347fde5ef0e7 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Tue, 29 Apr 2025 02:29:22 +0200
Subject: [PATCH 12/32] Delete DefsOrig.lean

---
 .../Geometry/Metric/Riemannian/DefsOrig.lean  | 271 ------------------
 1 file changed, 271 deletions(-)
 delete mode 100644 PhysLean/Mathematics/Geometry/Metric/Riemannian/DefsOrig.lean

diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/DefsOrig.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/DefsOrig.lean
deleted file mode 100644
index 410c850cd..000000000
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/DefsOrig.lean
+++ /dev/null
@@ -1,271 +0,0 @@
-/-
-Copyright (c) 2025 Matteo Cipollina. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Matteo Cipollina
--/
-
-import Mathlib.LinearAlgebra.QuadraticForm.Dual
-import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
-
-/-!
-# Riemannian Metrics on Manifolds
-
-/-!
-This file introduces `RiemannianMetric`, a smooth positive‐definite specialization of
-`PseudoRiemannianMetric` over ℝ.  It requires that each bilinear form `gₓ` satisfy
-`gₓ(v,v) > 0` for nonzero tangent vectors `v`, and uses this to define the canonical
-inner product, norm, and distance on each tangent space.  (TODO: assemble these
-data into an `InnerProductSpace ℝ (TangentSpace I_ℝ x)` instance).
--/
-
-## Main Definitions
-
-*   `RiemannianMetric I n M`: A structure representing a `C^n` Riemannian metric on a manifold `M`
-    modelled on `(E_ℝ, H_ℝ)` with model `I_ℝ`. This specializes `PseudoRiemannianMetric` to the
-    case where the base field is `ℝ` and requires the bilinear form `gₓ` at each point `x` to be
-    positive-definite.
-*   `RiemannianMetric.inner g x`: The inner product (bilinear form `gₓ`) on the tangent space `TₓM`.
-*   `RiemannianMetric.sharp g x`: The musical isomorphism (index raising) from the cotangent space
-    `T*ₓM` to the tangent space `TₓM`, which is an isomorphism in the Riemannian case due to
-    positive definiteness.
-*   `RiemannianMetric.toQuadraticForm g x`: Expresses the metric at point `x` as a quadratic form.
-
-## Implementation Notes
-
-*   Extends `PseudoRiemannianMetric` by fixing the base field to `ℝ` and adding the `pos_def'`
-    field, ensuring `gₓ(v, v) > 0` for non-zero `v`.
-*   The positive definiteness allows defining an `InnerProductSpace ℝ (TₓM)` instance on each
-    tangent space `TₓM`.
--/
-universe u v w
-
-open PseudoRiemannianMetric Bundle ContinuousLinearMap
-
-noncomputable section
-
-variable {E_ℝ : Type v} [NormedAddCommGroup E_ℝ] [NormedSpace ℝ E_ℝ] [FiniteDimensional ℝ E_ℝ]
-variable {H_ℝ : Type w} [TopologicalSpace H_ℝ]
-variable {M_ℝ : Type w} [TopologicalSpace M_ℝ] [ChartedSpace H_ℝ M_ℝ] [ChartedSpace H_ℝ E_ℝ]
-
-/-- A Riemannian metric of smoothness class `n` on a manifold `M` over `ℝ`.
-    This extends a pseudo-Riemannian metric with the positive definiteness condition. -/
-@[ext]
-structure RiemannianMetric
-  (I_ℝ : ModelWithCorners ℝ E_ℝ H_ℝ)
-  (n    : WithTop ℕ∞)
-  [inst_top : TopologicalSpace (TangentBundle I_ℝ M_ℝ)]
-  [inst_fib : FiberBundle      E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
-  [inst_vec : VectorBundle    ℝ E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
-  [inst_mani : IsManifold       I_ℝ (n + 1) M_ℝ]
-  [inst_cmvb : ContMDiffVectorBundle n E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _) I_ℝ]
-  extends @PseudoRiemannianMetric ℝ Real.instRCLike E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib
-    inst_vec inst_mani inst_cmvb where
-  /-- `gₓ(v, v) > 0` for all nonzero `v`. -/
-  pos_def' : ∀ x v, v ≠ 0 → val x v v > 0
-
-namespace RiemannianMetric
-
-variable {I_ℝ : ModelWithCorners ℝ E_ℝ H_ℝ} {n : WithTop ℕ∞}
-variable [inst_top : TopologicalSpace (TangentBundle I_ℝ M_ℝ)]
-variable [inst_fib : FiberBundle E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
-variable [inst_vec : VectorBundle ℝ E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _)]
-variable [inst_mani : IsManifold I_ℝ (n + 1) M_ℝ]
-variable [inst_cmvb : ContMDiffVectorBundle n E_ℝ (TangentSpace I_ℝ : M_ℝ → Type _) I_ℝ]
-
-/-- Coercion from RiemannianMetric to its underlying PseudoRiemannianMetric. -/
-instance : Coe (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
-      inst_cmvb)
-                (@PseudoRiemannianMetric ℝ Real.instRCLike E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top
-         inst_fib inst_vec inst_mani inst_cmvb) where
-  coe g := g.toPseudoRiemannianMetric
-
-omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
-@[simp] lemma pos_def (g : RiemannianMetric I_ℝ n) (x : M_ℝ) (v : TangentSpace I_ℝ x) (hv : v ≠ 0) :
-  (g.toPseudoRiemannianMetric.val x v) v > 0 := g.pos_def' x v hv
-
-/-- The inverse of the musical isomorphism (index raising), which is an isomorphism
-    in the Riemannian case. This is well-defined because the metric is positive definite. -/
-def sharp
-    (g : (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
-      inst_cmvb)) (x : M_ℝ) :
-    Module.Dual ℝ (TangentSpace I_ℝ x) ≃ₗ[ℝ] TangentSpace I_ℝ x :=
-  let bilin := g.toPseudoRiemannianMetric.toBilinForm x
-  have h_nondeg : bilin.Nondegenerate :=
-    g.toPseudoRiemannianMetric.toBilinForm_nondegenerate x
-  LinearEquiv.symm (bilin.toDual h_nondeg)
-
-/-- Express a Riemannian metric at a point as a quadratic form. -/
-abbrev toQuadraticForm
-    (g : (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
-      inst_cmvb))
-    (x : M_ℝ) :
-    QuadraticForm ℝ (TangentSpace I_ℝ x) :=
-  g.toPseudoRiemannianMetric.toQuadraticForm x
-
-omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
-/-- The quadratic form associated with a Riemannian metric is positive definite. -/
-lemma toQuadraticForm_posDef (g : RiemannianMetric I_ℝ n) (x : M_ℝ) :
-    (g.toQuadraticForm x).PosDef :=
-  λ v hv => g.pos_def x v hv
-
-omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
-/-- The application of a Riemannian metric's quadratic form to a vector. -/
-lemma toQuadraticForm_apply (g : RiemannianMetric I_ℝ n) (x : M_ℝ)
-    (v : TangentSpace I_ℝ x) :
-    g.toQuadraticForm x v = g.val x v v := by
-  simp only [toQuadraticForm, PseudoRiemannianMetric.toQuadraticForm_apply]
-
-omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
-/-- A positive definite quadratic form is nondegenerate. -/
-lemma posDef_to_nondegenerate [Invertible (2 : ℝ)]
-    (g : RiemannianMetric I_ℝ n) (x : M_ℝ) :
-    (QuadraticMap.associated (R := ℝ) (N := ℝ) (g.toQuadraticForm x)).Nondegenerate := by
-  let Q := g.toQuadraticForm x
-  let B := QuadraticMap.associated (R := ℝ) (N := ℝ) Q
-  constructor
-  · intro v h_all_zero_w
-    specialize h_all_zero_w v
-    have h_assoc_self : B v v = Q v :=
-        QuadraticMap.associated_eq_self_apply ℝ Q v
-    rw [h_assoc_self] at h_all_zero_w
-    by_cases h_v_zero : v = 0
-    · exact h_v_zero
-    · have h_quad_pos : Q v > 0 := g.pos_def x v h_v_zero
-      linarith [h_all_zero_w, h_quad_pos]
-  · intro w h_all_zero_v
-    specialize h_all_zero_v w
-    have h_assoc_self : B w w = Q w :=
-        QuadraticMap.associated_eq_self_apply ℝ Q w
-    rw [h_assoc_self] at h_all_zero_v
-    by_cases h_w_zero : w = 0
-    · exact h_w_zero
-    · have h_quad_pos : Q w > 0 := g.pos_def x w h_w_zero
-      linarith [h_all_zero_v, h_quad_pos]
-
-/-- The isometry between `(Q.prod <| -Q)` and `QuadraticForm.dualProd`,
-    where `Q` is the quadratic form associated with the Riemannian metric at `x`. -/
-abbrev toDualProdIso [Invertible (2 : ℝ)] (g : RiemannianMetric I_ℝ n) (x : M_ℝ) :
-    ((g.toQuadraticForm x).prod <| -(g.toQuadraticForm x)) →qᵢ
-    (QuadraticForm.dualProd ℝ (TangentSpace I_ℝ x)) :=
-  QuadraticForm.toDualProd (g.toQuadraticForm x)
-
-/-- The inner product on the tangent space at point `x` induced by the Riemannian metric `g`. -/
-def inner
-    (g : (@RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani
-      inst_cmvb)) (x : M_ℝ) (v w : TangentSpace I_ℝ x) : ℝ :=
-  g.toPseudoRiemannianMetric.val x v w
-
-omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
-@[simp] lemma inner_apply (g : RiemannianMetric I_ℝ n) (x : M_ℝ) (v w : TangentSpace I_ℝ x) :
-  inner g x v w = g.val x v w := rfl
-
-variable (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n _ _ _ _ _) (x : M_ℝ)
-
-omit [FiniteDimensional ℝ E_ℝ] [ChartedSpace H_ℝ E_ℝ] in
-/-- Pointwise symmetry of the inner product. -/
-lemma inner_symm (v w : TangentSpace I_ℝ x) :
-    g.inner x v w = g.inner x w v := by
-  simp only [inner_apply]
-  exact g.toPseudoRiemannianMetric.symm' x v w
-
-/-- The inner product space core for the tangent space at a point, derived from the
-Riemannian metric. -/
-noncomputable def tangentInnerCore
-  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
-  (x : M_ℝ) :
-  InnerProductSpace.Core ℝ (TangentSpace I_ℝ x) where
-  inner := λ v w => g.inner x v w
-  conj_symm := λ v w => by
-    simp only [inner_apply, conj_trivial]
-    exact g.toPseudoRiemannianMetric.symm' x w v
-  nonneg_re := λ v => by
-    simp only [inner_apply, RCLike.re_to_real]
-    by_cases hv : v = 0
-    · simp [hv, inner_apply, map_zero, zero_apply]
-    · exact le_of_lt (g.pos_def x v hv)
-  add_left := λ u v w => by
-    simp only [inner_apply, map_add, add_apply]
-  smul_left := λ r u v => by
-    simp only [inner_apply, map_smul, smul_apply, conj_trivial]
-    rfl
-  definite := fun v (h_inner_zero : g.inner x v v = 0) => by
-    by_contra h_v_ne_zero
-    have h_pos : g.inner x v v > 0 := g.pos_def x v h_v_ne_zero
-    linarith [h_inner_zero, h_pos]
-
-/-! ### Inner product space structure. -/
-
-/-- Normed additive commutative group structure on the tangent space at a point `x`,
-derived from the Riemannian metric `g`. -/
-noncomputable def TangentSpace.metricNormedAddCommGroup
-  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
-  (x : M_ℝ) :
-  NormedAddCommGroup (TangentSpace I_ℝ x) :=
-  @InnerProductSpace.Core.toNormedAddCommGroup ℝ (TangentSpace I_ℝ x) _ _ _ (tangentInnerCore g x)
-
-/-- The pre-inner product space core for the tangent space at a point,
-derived from the Riemannian metric. -/
-noncomputable def tangentPreInnerCore
-  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
-  (x : M_ℝ) :
-  PreInnerProductSpace.Core ℝ (TangentSpace I_ℝ x) where
-  inner := λ v w => g.inner x v w
-  conj_symm := λ v w => by
-    simp only [inner_apply, conj_trivial]
-    exact g.toPseudoRiemannianMetric.symm' x w v
-  nonneg_re := λ v => by
-    simp only [inner_apply, RCLike.re_to_real]
-    by_cases hv : v = 0
-    · simp [hv, inner_apply, map_zero, zero_apply]
-    · exact le_of_lt (g.pos_def x v hv)
-  add_left := λ u v w => by
-    simp only [inner_apply, map_add, add_apply]
-  smul_left := λ r u v => by
-    simp only [inner_apply, map_smul, smul_apply, conj_trivial]
-    rfl
-
-/-- Each tangent space carries a seminormed add comm group structure derived from
-the Riemannian metric -/
-noncomputable def TangentSpace.metricSeminormedAddCommGroup
-  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
-  (x : M_ℝ) :
-  SeminormedAddCommGroup (TangentSpace I_ℝ x) :=
-  (TangentSpace.metricNormedAddCommGroup g x).toSeminormedAddCommGroup
-/-
-noncomputable def TangentSpace.metricInnerProductSpace
-  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
-  (x : M_ℝ) :
-  InnerProductSpace ℝ (TangentSpace I_ℝ x) := by
-
-  -- Ensure the underlying group structure is known
-  letI : AddCommGroup (TangentSpace I_ℝ x) := TangentSpace.addCommGroup x
-  letI : TopologicalSpace (TangentSpace I_ℝ x) := inferInstance
-  letI : TopologicalAddGroup (TangentSpace I_ℝ x) := inferInstance
-  letI : UniformSpace (TangentSpace I_ℝ x) := inferInstance
-  letI : UniformAddGroup (TangentSpace I_ℝ x) := ⟨⟩
-
-  -- Provide the specific SeminormedAddCommGroup instance
-  letI : SeminormedAddCommGroup (TangentSpace I_ℝ x) :=
-    TangentSpace.metricSeminormedAddCommGroup g x
-
-  -- Provide the NormedAddCommGroup instance
-  letI : NormedAddCommGroup (TangentSpace I_ℝ x) :=
-    TangentSpace.metricNormedAddCommGroup g x
-
-  -- Build the InnerProductSpace using the custom inner core
-  let core := tangentInnerCore g x
-  letI : NormedSpace ℝ (TangentSpace I_ℝ x) :=
-    @InnerProductSpace.Core.toNormedSpace ℝ (TangentSpace I_ℝ x) _ _ _ core
-  exact InnerProductSpace.ofCore core
-
-
--- Check the norm_sq_eq_inner property holds for the constructed structure
-lemma TangentSpace.metric_norm_sq_eq_inner
-  (g : @RiemannianMetric E_ℝ _ _ H_ℝ _ M_ℝ _ _ I_ℝ n inst_top inst_fib inst_vec inst_mani inst_cmvb)
-  (x : M_ℝ) (v : TangentSpace I_ℝ x) :
-  letI := TangentSpace.metricNormedAddCommGroup g x
-  ‖v‖ ^ 2 = g.inner x v v := by
-  letI := TangentSpace.metricInnerProductSpace g x
-  exact InnerProductSpace.norm_sq_eq_inner v
--/
-end RiemannianMetric

From 456c233641f2a096f252383650afe6e494789a96 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Tue, 29 Apr 2025 02:41:51 +0200
Subject: [PATCH 13/32] fix

---
 .../Metric/PseudoRiemannian/Defs.lean         | 46 +++++++++----------
 .../Geometry/Metric/Riemannian/Defs.lean      |  7 +++
 2 files changed, 29 insertions(+), 24 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index 22d87d9f9..dfab1dea6 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -58,14 +58,12 @@ variable {K : Type*} [Field K] [LinearOrder K]
     of a maximal negative definite subspace. -/
 noncomputable def negDim {E : Type*} [AddCommGroup E]
     [Module ℝ E] [FiniteDimensional ℝ E]
-    (q : QuadraticForm ℝ E) : ℕ :=
-  by
-    classical
-    let P : (Fin (finrank ℝ E) → SignType) → Prop := fun w =>
-        QuadraticMap.Equivalent q (QuadraticMap.weightedSumSquares ℝ fun i => (w i : ℝ))
-    let h_exists : ∃ w, P w := QuadraticForm.equivalent_signType_weighted_sum_squared q
-    let w := Classical.choose h_exists
-    exact Finset.card (Finset.filter (fun i => w i = SignType.neg) Finset.univ)
+    (q : QuadraticForm ℝ E) : ℕ := by classical
+  let P : (Fin (finrank ℝ E) → SignType) → Prop := fun w =>
+      QuadraticMap.Equivalent q (QuadraticMap.weightedSumSquares ℝ fun i => (w i : ℝ))
+  let h_exists : ∃ w, P w := QuadraticForm.equivalent_signType_weighted_sum_squared q
+  let w := Classical.choose h_exists
+  exact Finset.card (Finset.filter (fun i => w i = SignType.neg) Finset.univ)
 
 /-- The i-th standard basis vector has a 1 in the i-th position. -/
 lemma Pi.basisFun_apply_same {𝕜 ι : Type*} [Field 𝕜] [Fintype ι] [DecidableEq ι] (i : ι) :
@@ -290,12 +288,12 @@ lemma symm' (x : M) (v w : TangentSpace I x) : (g.val x v) w = (g.val x w) v :=
   g.symm x v w
 
 lemma nondegenerate' (x : M) (v : TangentSpace I x)
-   (h : ∀ w : TangentSpace I x, (g.val x v) w = 0) :
-  v = 0 :=
+    (h : ∀ w : TangentSpace I x, (g.val x v) w = 0) :
+    v = 0 :=
   g.nondegenerate x v h
 
 lemma smooth' (x₀ : M) (v w : E) :
-  ContDiffWithinAt ℝ n (fun y => g.val ((extChartAt I x₀).symm y)
+    ContDiffWithinAt ℝ n (fun y => g.val ((extChartAt I x₀).symm y)
     (mfderiv I I ((extChartAt I x₀).symm) y v) (mfderiv I I ((extChartAt I x₀).symm) y w))
     ((extChartAt I x₀).target) ((extChartAt I x₀) x₀) :=
   g.smooth_in_charts' x₀ v w
@@ -406,37 +404,37 @@ lemma finrank_continuousDual_eq_finrank {𝕜 E : Type*} [NontriviallyNormedFiel
       ⟨g.flatL_inj x, g.flatL_surj x⟩)
 
 lemma coe_flatEquiv
-     (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) :
     (g.flatEquiv x : TangentSpace I x →ₗ[ℝ] (TangentSpace I x →L[ℝ] ℝ)) = g.flatL x := rfl
 
 @[simp] lemma flatEquiv_apply
-     (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
     (g.flatEquiv x v) w = g.val x v w := rfl
 
 /-- The "musical" isomorphism (index raising) from the dual of the tangent space to the
     tangent space, induced by a pseudo-Riemannian metric. This is the inverse of `flatEquiv`. -/
 def sharpEquiv
-     (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) :
     (TangentSpace I x →L[ℝ] ℝ) ≃L[ℝ] TangentSpace I x :=
   (g.flatEquiv x).symm
 
 /-- The index raising map `sharp` as a continuous linear map. -/
 def sharpL
-     (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) :
     (TangentSpace I x →L[ℝ] ℝ) →L[ℝ] TangentSpace I x :=
   (g.sharpEquiv x).toContinuousLinearMap
 
 lemma sharpL_eq_toContinuousLinearMap
-     (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) :
   g.sharpL x = (g.sharpEquiv x).toContinuousLinearMap := rfl
 
 lemma coe_sharpEquiv
-     (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) :
     (g.sharpEquiv x : (TangentSpace I x →L[ℝ] ℝ) →L[ℝ] TangentSpace I x) = g.sharpL x := rfl
 
 /-- The index raising map `sharp` as a linear map. -/
 noncomputable def sharp
-     (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) :
     (TangentSpace I x →L[ℝ] ℝ) →ₗ[ℝ] TangentSpace I x :=
   (g.sharpEquiv x).toLinearEquiv.toLinearMap
 
@@ -446,13 +444,13 @@ noncomputable def sharp
   (g.flatEquiv x).left_inv v
 
 @[simp] lemma flatL_apply_sharpL
-     (g : PseudoRiemannianMetric E H M n I) (x : M) (ω : TangentSpace I x →L[ℝ] ℝ) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) (ω : TangentSpace I x →L[ℝ] ℝ) :
     g.flatL x (g.sharpL x ω) = ω :=
   (g.flatEquiv x).right_inv ω
 
 /-- Applying `sharp` then `flat` recovers the original covector. -/
 @[simp] lemma flat_sharp_apply
-     (g : PseudoRiemannianMetric E H M n I) (x : M) (ω : TangentSpace I x →L[ℝ] ℝ) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) (ω : TangentSpace I x →L[ℝ] ℝ) :
     g.flat x (g.sharp x ω) = ω := by
   have := flatL_apply_sharpL g x ω
   simp only [sharp, sharpL, flat, flatL, coe_flatEquiv]; simp only [coe_sharpEquiv,
@@ -460,7 +458,7 @@ noncomputable def sharp
   exact this
 
 @[simp] lemma sharp_flat_apply
-     (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) :
     g.sharp x (g.flat x v) = v := by
   have := sharpL_apply_flatL g x v
   simp only [sharp, sharpL, flat, flatL]; simp only [coe_flatEquiv, coe_sharpEquiv,
@@ -469,15 +467,15 @@ noncomputable def sharp
 
 /-- The metric evaluated at `sharp ω₁` and `sharp ω₂`. -/
 @[simp] lemma apply_sharp_sharp
-     (g : PseudoRiemannianMetric E H M n I) (x : M) (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
     g.val x (g.sharpL x ω₁) (g.sharpL x ω₂) = ω₁ (g.sharpL x ω₂) := by
   rw [← flatL_apply g x (g.sharpL x ω₁)]
   rw [flatL_apply_sharpL g x ω₁]
 
 /-- The metric evaluated at `v` and `sharp ω`. -/
 lemma apply_vec_sharp
-     (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x)
-     (ω : TangentSpace I x →L[ℝ] ℝ) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x)
+    (ω : TangentSpace I x →L[ℝ] ℝ) :
     g.val x v (g.sharpL x ω) = ω v := by
   rw [g.symm' x v (g.sharpL x ω)]
   rw [← flatL_apply g x (g.sharpL x ω)]
diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index 30bdd2405..0dfab583f 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -1,5 +1,12 @@
+/-
+Copyright (c) 2025 Matteo Cipollina. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Matteo Cipollina
+-/
+
 import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
 import Mathlib.LinearAlgebra.QuadraticForm.Dual
+
 section RiemannianMetric
 
 open Bundle Set Finset Function Filter Module Topology ContinuousLinearMap

From 8df93c2f04115ac6e45f6803fd29745d68f588c8 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Tue, 29 Apr 2025 02:55:05 +0200
Subject: [PATCH 14/32] fix lints

---
 .../Metric/PseudoRiemannian/Defs.lean         | 21 +++++++------------
 .../Geometry/Metric/Riemannian/Defs.lean      |  6 ++++++
 2 files changed, 13 insertions(+), 14 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index dfab1dea6..e6db5149e 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -67,13 +67,12 @@ noncomputable def negDim {E : Type*} [AddCommGroup E]
 
 /-- The i-th standard basis vector has a 1 in the i-th position. -/
 lemma Pi.basisFun_apply_same {𝕜 ι : Type*} [Field 𝕜] [Fintype ι] [DecidableEq ι] (i : ι) :
-  (Pi.basisFun 𝕜 ι) i i = 1 := by
+    (Pi.basisFun 𝕜 ι) i i = 1 := by
   simp only [Pi.basisFun_apply, Pi.single_eq_same]
 
 /-- The i-th standard basis vector has 0 in all positions j ≠ i. -/
 lemma Pi.basisFun_apply_ne {𝕜 ι : Type*} [Field 𝕜] [Fintype ι] [DecidableEq ι] (i j : ι)
-  (h : j ≠ i) :
-  (Pi.basisFun 𝕜 ι) i j = 0 := by
+    (h : j ≠ i) : (Pi.basisFun 𝕜 ι) i j = 0 := by
   simp only [Pi.basisFun_apply, Pi.single_eq_of_ne h]
 
 /-- For a standard basis vector in a weighted sum of squares, only one term in the sum
@@ -288,15 +287,12 @@ lemma symm' (x : M) (v w : TangentSpace I x) : (g.val x v) w = (g.val x w) v :=
   g.symm x v w
 
 lemma nondegenerate' (x : M) (v : TangentSpace I x)
-    (h : ∀ w : TangentSpace I x, (g.val x v) w = 0) :
-    v = 0 :=
-  g.nondegenerate x v h
+    (h : ∀ w : TangentSpace I x, (g.val x v) w = 0) : v = 0 := g.nondegenerate x v h
 
 lemma smooth' (x₀ : M) (v w : E) :
     ContDiffWithinAt ℝ n (fun y => g.val ((extChartAt I x₀).symm y)
     (mfderiv I I ((extChartAt I x₀).symm) y v) (mfderiv I I ((extChartAt I x₀).symm) y w))
-    ((extChartAt I x₀).target) ((extChartAt I x₀) x₀) :=
-  g.smooth_in_charts' x₀ v w
+    ((extChartAt I x₀).target) ((extChartAt I x₀) x₀) := g.smooth_in_charts' x₀ v w
 
 lemma negDim_isLocallyConstant' : IsLocallyConstant (fun x => (toQuadraticForm g x).negDim) :=
   g.negDim_isLocallyConstant
@@ -421,8 +417,7 @@ def sharpEquiv
 /-- The index raising map `sharp` as a continuous linear map. -/
 def sharpL
     (g : PseudoRiemannianMetric E H M n I) (x : M) :
-    (TangentSpace I x →L[ℝ] ℝ) →L[ℝ] TangentSpace I x :=
-  (g.sharpEquiv x).toContinuousLinearMap
+    (TangentSpace I x →L[ℝ] ℝ) →L[ℝ] TangentSpace I x := (g.sharpEquiv x).toContinuousLinearMap
 
 lemma sharpL_eq_toContinuousLinearMap
     (g : PseudoRiemannianMetric E H M n I) (x : M) :
@@ -435,8 +430,7 @@ lemma coe_sharpEquiv
 /-- The index raising map `sharp` as a linear map. -/
 noncomputable def sharp
     (g : PseudoRiemannianMetric E H M n I) (x : M) :
-    (TangentSpace I x →L[ℝ] ℝ) →ₗ[ℝ] TangentSpace I x :=
-  (g.sharpEquiv x).toLinearEquiv.toLinearMap
+    (TangentSpace I x →L[ℝ] ℝ) →ₗ[ℝ] TangentSpace I x := (g.sharpEquiv x).toLinearEquiv.toLinearMap
 
 @[simp] lemma sharpL_apply_flatL
      (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) :
@@ -445,8 +439,7 @@ noncomputable def sharp
 
 @[simp] lemma flatL_apply_sharpL
     (g : PseudoRiemannianMetric E H M n I) (x : M) (ω : TangentSpace I x →L[ℝ] ℝ) :
-    g.flatL x (g.sharpL x ω) = ω :=
-  (g.flatEquiv x).right_inv ω
+    g.flatL x (g.sharpL x ω) = ω := (g.flatEquiv x).right_inv ω
 
 /-- Applying `sharp` then `flat` recovers the original covector. -/
 @[simp] lemma flat_sharp_apply
diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index 0dfab583f..4eda3f7ba 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -7,6 +7,12 @@ Authors: Matteo Cipollina
 import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
 import Mathlib.LinearAlgebra.QuadraticForm.Dual
 
+/-!
+# Riemannian Metric Definitions
+
+This module defines the Riemannian metric, building on pseudo-Riemannian metrics.
+-/
+namespace PseudoRiemannianMetric
 section RiemannianMetric
 
 open Bundle Set Finset Function Filter Module Topology ContinuousLinearMap

From 95e8d3a739c9fb559ca6d4d692ec314dd03eea26 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Tue, 29 Apr 2025 14:41:11 +0200
Subject: [PATCH 15/32] fix lints

---
 .../Metric/PseudoRiemannian/Defs.lean         | 123 ++++++++++++++++++
 .../Geometry/Metric/Riemannian/Defs.lean      |  36 ++++-
 2 files changed, 156 insertions(+), 3 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index e6db5149e..a020f5289 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -475,3 +475,126 @@ lemma apply_vec_sharp
   rw [flatL_apply_sharpL g x ω]
 
 end PseudoRiemannianMetric
+
+namespace PseudoRiemannianMetric
+
+variable {E : Type v} {H : Type w} {M : Type w} {n : WithTop ℕ∞}
+variable [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+variable [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E]
+variable {I : ModelWithCorners ℝ E H}
+variable [IsManifold I (n + 1) M]
+variable [TopologicalSpace (TotalSpace E (TangentSpace I : M → Type _))]
+variable [FiberBundle E (TangentSpace I : M → Type _)]
+variable [VectorBundle ℝ E (TangentSpace I : M → Type _)]
+variable [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
+variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
+
+section Cotangent
+
+/-- The value of the induced metric on the cotangent space at point `x`. -/
+noncomputable def cotangentMetricVal (g : PseudoRiemannianMetric E H M n I) (x : M)
+    (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) : ℝ :=
+  g.val x (g.sharpL x ω₁) (g.sharpL x ω₂)
+
+@[simp] lemma cotangentMetricVal_eq_apply_sharp (g : PseudoRiemannianMetric E H M n I) (x : M)
+    (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
+  cotangentMetricVal g x ω₁ ω₂ = ω₁ (g.sharpL x ω₂) := by
+  rw [cotangentMetricVal, apply_sharp_sharp]
+
+lemma cotangentMetricVal_symm (g : PseudoRiemannianMetric E H M n I) (x : M)
+    (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
+  cotangentMetricVal g x ω₁ ω₂ = cotangentMetricVal g x ω₂ ω₁ := by
+  unfold cotangentMetricVal
+  rw [g.symm' x (g.sharpL x ω₁) (g.sharpL x ω₂)]
+
+/-- The induced metric on the cotangent space at point `x` as a bilinear form. -/
+noncomputable def cotangentToBilinForm (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    LinearMap.BilinForm ℝ (TangentSpace I x →L[ℝ] ℝ) where
+  toFun ω₁ := { toFun := λ ω₂ => cotangentMetricVal g x ω₁ ω₂,
+                map_add' := λ ω₂ ω₃ => by
+                  simp only [cotangentMetricVal,
+                    ContinuousLinearMap.map_add,
+                    ContinuousLinearMap.add_apply],
+                map_smul' := λ c ω₂ => by
+                  simp only [cotangentMetricVal,
+                    map_smul, smul_eq_mul, RingHom.id_apply] }
+  map_add' := λ ω₁ ω₂ => by
+    ext ω₃
+    simp only [cotangentMetricVal,
+      ContinuousLinearMap.map_add,
+      ContinuousLinearMap.add_apply,
+      LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
+  map_smul' := λ c ω₁ => by
+    ext ω₂
+    simp only [cotangentMetricVal,
+      ContinuousLinearMap.map_smul,
+      ContinuousLinearMap.smul_apply,
+      LinearMap.coe_mk, AddHom.coe_mk,
+      RingHom.id_apply, LinearMap.smul_apply]
+
+/-- The induced metric on the cotangent space at point `x` as a quadratic form. -/
+noncomputable def cotangentToQuadraticForm (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    QuadraticForm ℝ (TangentSpace I x →L[ℝ] ℝ) where
+  toFun ω := cotangentMetricVal g x ω ω
+  toFun_smul a ω := by
+    simp only [cotangentMetricVal,
+      ContinuousLinearMap.map_smul,
+      ContinuousLinearMap.smul_apply,
+      smul_smul, pow_two]
+  exists_companion' :=
+      ⟨ LinearMap.mk₂ ℝ (fun ω₁ ω₂ =>
+          cotangentMetricVal g x ω₁ ω₂ + cotangentMetricVal g x ω₂ ω₁)
+        (fun ω₁ ω₂ ω₃ => by simp only [cotangentMetricVal, map_add, add_apply]; ring)
+        (fun a ω₁ ω₂ => by
+          simp only [cotangentMetricVal, map_smul, smul_apply];
+          ring_nf; exact Eq.symm (smul_add ..))
+        (fun ω₁ ω₂ ω₃ => by
+          simp only [cotangentMetricVal, map_add, add_apply]; ring)
+        (fun a ω₁ ω₂ => by
+          simp only [cotangentMetricVal, map_smul, smul_apply]; ring_nf;
+          exact Eq.symm (smul_add ..)),
+          by
+            intro ω₁ ω₂
+            simp only [LinearMap.mk₂_apply, cotangentMetricVal]
+            simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.add_apply]
+            ring⟩
+
+@[simp] lemma cotangentToBilinForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M)
+    (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
+  cotangentToBilinForm g x ω₁ ω₂ = cotangentMetricVal g x ω₁ ω₂ := rfl
+
+@[simp] lemma cotangentToQuadraticForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M)
+    (ω : TangentSpace I x →L[ℝ] ℝ) :
+  cotangentToQuadraticForm g x ω = cotangentMetricVal g x ω ω := rfl
+
+@[simp] lemma cotangentToBilinForm_isSymm (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (cotangentToBilinForm g x).IsSymm := by
+  intro ω₁ ω₂; simp only [cotangentToBilinForm_apply]; exact cotangentMetricVal_symm g x ω₁ ω₂
+
+/-- The cotangent metric is non-degenerate: if `cotangentMetricVal g x ω v = 0` for all `v`,
+    then `ω = 0`. -/
+lemma cotangentMetricVal_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x : M)
+    (ω : TangentSpace I x →L[ℝ] ℝ) (h : ∀ v : TangentSpace I x →L[ℝ] ℝ,
+      cotangentMetricVal g x ω v = 0) :
+    ω = 0 := by
+  apply ContinuousLinearMap.ext
+  intro v
+  have h_forall : ∀ w : TangentSpace I x, ω w = 0 := by
+    intro w
+    let ω' : TangentSpace I x →L[ℝ] ℝ := g.flatL x w
+    have this : g.sharpL x ω' = w := by
+      simp only [ω', sharpL_apply_flatL]
+    have h_apply : cotangentMetricVal g x ω ω' = 0 := h ω'
+    simp only [cotangentMetricVal_eq_apply_sharp] at h_apply
+    rw [this] at h_apply
+    exact h_apply
+  exact h_forall v
+
+@[simp] lemma cotangentToBilinForm_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x : M) :
+    (cotangentToBilinForm g x).Nondegenerate := by
+  intro ω hω
+  apply cotangentMetricVal_nondegenerate g x ω
+  intro v
+  exact hω v
+
+end Cotangent
diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index 4eda3f7ba..d3179c311 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -7,6 +7,20 @@ Authors: Matteo Cipollina
 import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
 import Mathlib.LinearAlgebra.QuadraticForm.Dual
 
+import Mathlib.Algebra.Lie.OfAssociative
+import Mathlib.Analysis.InnerProductSpace.Basic
+import Mathlib.Analysis.Normed.Field.Instances
+import Mathlib.Analysis.RCLike.Lemmas
+import Mathlib.Data.Real.StarOrdered
+import Mathlib.Geometry.Manifold.MFDeriv.Defs
+import Mathlib.Geometry.Manifold.VectorBundle.Basic
+import Mathlib.LinearAlgebra.BilinearForm.Properties
+import Mathlib.LinearAlgebra.FreeModule.PID
+import Mathlib.LinearAlgebra.QuadraticForm.Real
+import Mathlib.RingTheory.Henselian
+import Mathlib.Topology.Algebra.Module.ModuleTopology
+import Mathlib.Topology.LocallyConstant.Basic
+import Mathlib.MeasureTheory.Integral.IntervalIntegral
 /-!
 # Riemannian Metric Definitions
 
@@ -114,12 +128,12 @@ noncomputable def tangentInnerCore (g : RiemannianMetric I n M) (x : M) :
   nonneg_re := λ v => by
     simp only [inner_apply, RCLike.re_to_real]
     by_cases hv : v = 0
-    · simp [hv, inner_apply, map_zero, zero_apply]
+    · simp [hv, inner_apply, map_zero]
     · exact le_of_lt (g.pos_def x v hv)
   add_left := λ u v w => by
-    simp only [inner_apply, map_add, add_apply]
+    simp only [inner_apply, map_add, ContinuousLinearMap.add_apply]
   smul_left := λ r u v => by
-    simp only [inner_apply, map_smul, smul_apply, conj_trivial]
+    simp only [inner_apply, map_smul, conj_trivial]
     rfl
   definite := fun v (h_inner_zero : g.inner x v v = 0) => by
     by_contra h_v_ne_zero
@@ -186,4 +200,20 @@ example (g : RiemannianMetric I n M) (x : M) (v : TangentSpace I x) : ℝ :=
   let normed_group := TangentSpace.metricNormedAddCommGroup g x
   @Norm.norm (TangentSpace I x) (@NormedAddCommGroup.toNorm (TangentSpace I x) normed_group) v
 
+lemma norm_eq_norm_of_metricNormedAddCommGroup (g : RiemannianMetric I n M) (x : M)
+    (v : TangentSpace I x) : norm g x v = @Norm.norm (TangentSpace I x)
+    (@NormedAddCommGroup.toNorm _ (TangentSpace.metricNormedAddCommGroup g x)) v := by
+  unfold norm
+  let normed_group := TangentSpace.metricNormedAddCommGroup g x
+  unfold TangentSpace.metricNormedAddCommGroup
+  simp only [inner_apply]
+  rfl
+
+/-- Calculates the length of a curve `γ` between parameters `t₀` and `t₁`
+    using the Riemannian metric `g`. This is defined as the integral of the norm of
+    the tangent vector along the curve. -/
+def curveLength (g : RiemannianMetric I n M) (γ : ℝ → M) (t₀ t₁ : ℝ)
+    {IDE : ModelWithCorners ℝ ℝ ℝ}[ChartedSpace ℝ ℝ] : ℝ :=
+  ∫ t in t₀..t₁, norm g (γ t) ((mfderiv IDE I γ t) ((1 : ℝ) : TangentSpace IDE t))
+
 end RiemannianMetric

From 51cd77a62121c6739159eae632cf81c2fa8d920f Mon Sep 17 00:00:00 2001
From: jstoobysmith <72603918+jstoobysmith@users.noreply.github.com>
Date: Tue, 29 Apr 2025 16:02:21 +0000
Subject: [PATCH 16/32] feat: Remove unnecessary instances.

---
 .../Metric/PseudoRiemannian/Defs.lean         | 23 +++--------------
 .../Geometry/Metric/Riemannian/Defs.lean      | 25 +------------------
 .../HarmonicOscillator/Completeness.lean      |  2 +-
 3 files changed, 6 insertions(+), 44 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index a020f5289..ec3424649 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -3,15 +3,12 @@ Copyright (c) 2025 Matteo Cipollina. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Matteo Cipollina
 -/
-
 import Mathlib.Analysis.InnerProductSpace.Basic
 import Mathlib.Analysis.RCLike.Lemmas
 import Mathlib.Geometry.Manifold.MFDeriv.Defs
-import Mathlib.Geometry.Manifold.VectorBundle.Basic
 import Mathlib.LinearAlgebra.BilinearForm.Properties
 import Mathlib.LinearAlgebra.QuadraticForm.Real
 import Mathlib.Topology.LocallyConstant.Basic
-
 /-!
 # Pseudo-Riemannian Metrics on Smooth Manifolds
 
@@ -52,7 +49,7 @@ open scoped Manifold Bundle LinearMap Dual
 
 namespace QuadraticForm
 
-variable {K : Type*} [Field K] [LinearOrder K]
+variable {K : Type*} [Field K]
 
 /-- The negative dimension (or index) of a quadratic form is the dimension
     of a maximal negative definite subspace. -/
@@ -185,17 +182,12 @@ structure PseudoRiemannianMetric
     (E : Type v) (H : Type w) (M : Type w) (n : WithTop ℕ∞)
     [inst_norm_grp_E : NormedAddCommGroup E]
     [inst_norm_sp_E : NormedSpace ℝ E]
-    [inst_findim_E : FiniteDimensional ℝ E]
     [inst_top_H : TopologicalSpace H]
     [inst_top_M : TopologicalSpace M]
     [inst_chart_M : ChartedSpace H M]
     [inst_chart_E : ChartedSpace H E]
     (I : ModelWithCorners ℝ E H)
     [inst_mani : IsManifold I (n + 1) M]
-    [inst_total_top : TopologicalSpace (TotalSpace E (TangentSpace I : M → Type _))]
-    [inst_fb : FiberBundle E (TangentSpace I : M → Type _)]
-    [inst_vb : VectorBundle ℝ E (TangentSpace I : M → Type _)]
-    [inst_cmvb : ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
     [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)] :
       Type (max u v w) where
   /-- The metric tensor at each point `x : M`, represented as a continuous linear map
@@ -224,14 +216,10 @@ structure PseudoRiemannianMetric
 namespace PseudoRiemannianMetric
 
 variable {E : Type v} {H : Type w} {M : Type w} {n : WithTop ℕ∞}
-variable [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+variable [NormedAddCommGroup E] [NormedSpace ℝ E]
 variable [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E]
 variable {I : ModelWithCorners ℝ E H}
 variable [IsManifold I (n + 1) M]
-variable [TopologicalSpace (TotalSpace E (TangentSpace I : M → Type _))]
-variable [FiberBundle E (TangentSpace I : M → Type _)]
-variable [VectorBundle ℝ E (TangentSpace I : M → Type _)]
-variable [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
 variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
 variable {g : PseudoRiemannianMetric E H M n I}
 
@@ -479,14 +467,10 @@ end PseudoRiemannianMetric
 namespace PseudoRiemannianMetric
 
 variable {E : Type v} {H : Type w} {M : Type w} {n : WithTop ℕ∞}
-variable [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+variable [NormedAddCommGroup E] [NormedSpace ℝ E]
 variable [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E]
 variable {I : ModelWithCorners ℝ E H}
 variable [IsManifold I (n + 1) M]
-variable [TopologicalSpace (TotalSpace E (TangentSpace I : M → Type _))]
-variable [FiberBundle E (TangentSpace I : M → Type _)]
-variable [VectorBundle ℝ E (TangentSpace I : M → Type _)]
-variable [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
 variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
 
 section Cotangent
@@ -598,3 +582,4 @@ lemma cotangentMetricVal_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x
   exact hω v
 
 end Cotangent
+#min_imports
diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index d3179c311..7ecc6a160 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -3,24 +3,9 @@ Copyright (c) 2025 Matteo Cipollina. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Matteo Cipollina
 -/
-
-import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
-import Mathlib.LinearAlgebra.QuadraticForm.Dual
-
 import Mathlib.Algebra.Lie.OfAssociative
-import Mathlib.Analysis.InnerProductSpace.Basic
-import Mathlib.Analysis.Normed.Field.Instances
-import Mathlib.Analysis.RCLike.Lemmas
-import Mathlib.Data.Real.StarOrdered
-import Mathlib.Geometry.Manifold.MFDeriv.Defs
-import Mathlib.Geometry.Manifold.VectorBundle.Basic
-import Mathlib.LinearAlgebra.BilinearForm.Properties
-import Mathlib.LinearAlgebra.FreeModule.PID
-import Mathlib.LinearAlgebra.QuadraticForm.Real
-import Mathlib.RingTheory.Henselian
-import Mathlib.Topology.Algebra.Module.ModuleTopology
-import Mathlib.Topology.LocallyConstant.Basic
 import Mathlib.MeasureTheory.Integral.IntervalIntegral
+import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
 /-!
 # Riemannian Metric Definitions
 
@@ -46,10 +31,6 @@ variable {I : ModelWithCorners ℝ E H} {n : ℕ∞}
 structure RiemannianMetric
   (I : ModelWithCorners ℝ E H) (n : ℕ∞) (M : Type w)
   [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (n + 1) M]
-  [inst_top : TopologicalSpace (TangentBundle I M)]
-  [inst_fib : FiberBundle E (TangentSpace I : M → Type _)]
-  [inst_vec : VectorBundle ℝ E (TangentSpace I : M → Type _)]
-  [inst_cmvb : ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
   [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)] : Type _ where
   /-- The underlying pseudo-Riemannian metric. -/
   toPseudoRiemannianMetric : PseudoRiemannianMetric E H M (n) I
@@ -60,10 +41,6 @@ namespace RiemannianMetric
 
 variable {I : ModelWithCorners ℝ E H} {n : ℕ∞} {M : Type w}
 variable [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (n + 1) M]
-variable [inst_top : TopologicalSpace (TangentBundle I M)]
-variable [inst_fib : FiberBundle E (TangentSpace I : M → Type _)]
-variable [inst_vec : VectorBundle ℝ E (TangentSpace I : M → Type _)]
-variable [inst_cmvb : ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
 variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
 
 /-- Coercion from RiemannianMetric to its underlying PseudoRiemannianMetric. -/
diff --git a/PhysLean/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean b/PhysLean/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean
index 284fe6327..cc3c777ee 100644
--- a/PhysLean/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean
+++ b/PhysLean/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean
@@ -130,7 +130,7 @@ lemma mul_power_integrable (f : ℝ → ℂ) (hf : MemHS f) (r : ℕ) :
     cases h with
     | intro h =>
       have h' : Q.ξ = 0 := by
-        simpa [one_div] using h
+        simp [one_div] at h
       exact Q.ξ_pos.ne' h'
 
   · simp only [ne_eq, Decidable.not_not] at hr

From 57c311f0cbfd8d4006a46482412ef8cc5d92523f Mon Sep 17 00:00:00 2001
From: jstoobysmith <72603918+jstoobysmith@users.noreply.github.com>
Date: Tue, 29 Apr 2025 16:02:36 +0000
Subject: [PATCH 17/32] Update Defs.lean

---
 PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean | 1 -
 1 file changed, 1 deletion(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index ec3424649..5d081cfd6 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -582,4 +582,3 @@ lemma cotangentMetricVal_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x
   exact hω v
 
 end Cotangent
-#min_imports

From 139111301e2280e4d1e5dcfaa8a64481b7ce4336 Mon Sep 17 00:00:00 2001
From: jstoobysmith <72603918+jstoobysmith@users.noreply.github.com>
Date: Wed, 30 Apr 2025 05:13:44 +0000
Subject: [PATCH 18/32] refactor: Fix lint I broke

---
 PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index 7ecc6a160..d465bfa71 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -20,7 +20,7 @@ open PseudoRiemannianMetric InnerProductSpace
 
 noncomputable section
 
-variable {E : Type v} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+variable {E : Type v} [NormedAddCommGroup E] [NormedSpace ℝ E]
 variable {H : Type w} [TopologicalSpace H]
 variable {M : Type w} [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E]
 variable {I : ModelWithCorners ℝ E H} {n : ℕ∞}

From 55654d0d9baff57237bfc16a83f10d313a99ea5c Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Wed, 30 Apr 2025 18:17:59 +0200
Subject: [PATCH 19/32] Revert "merge"

---
 PhysLean.lean                                 |   1 -
 .../TightBindingChain/Basic.lean              |  24 ----
 .../StringTheory/FTheory/SU5U1/Charges.lean   |  12 --
 .../StringTheory/FTheory/SU5U1/Examples.lean  |  26 +---
 .../StringTheory/FTheory/SU5U1/Matter.lean    |  66 ---------
 .../FTheory/SU5U1/NoExotics/FiveBar.lean      |  15 +-
 .../FTheory/SU5U1/PhenoConstraints/Basic.lean | 135 ++++--------------
 7 files changed, 37 insertions(+), 242 deletions(-)
 delete mode 100644 PhysLean/CondensedMatter/TightBindingChain/Basic.lean

diff --git a/PhysLean.lean b/PhysLean.lean
index 1b95f4dc2..f72a1ba8e 100644
--- a/PhysLean.lean
+++ b/PhysLean.lean
@@ -7,7 +7,6 @@ import PhysLean.ClassicalMechanics.Space.VectorIdentities
 import PhysLean.ClassicalMechanics.Time.Basic
 import PhysLean.ClassicalMechanics.VectorFields
 import PhysLean.CondensedMatter.Basic
-import PhysLean.CondensedMatter.TightBindingChain.Basic
 import PhysLean.Cosmology.Basic
 import PhysLean.Cosmology.FLRW.Basic
 import PhysLean.Electromagnetism.Basic
diff --git a/PhysLean/CondensedMatter/TightBindingChain/Basic.lean b/PhysLean/CondensedMatter/TightBindingChain/Basic.lean
deleted file mode 100644
index ee8670625..000000000
--- a/PhysLean/CondensedMatter/TightBindingChain/Basic.lean
+++ /dev/null
@@ -1,24 +0,0 @@
-/-
-Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Joseph Tooby-Smith
--/
-import PhysLean.Meta.TODO.Basic
-/-!
-
-# The tight binding chain
-
-The tight binding chain corresponds to an electron in motion
-in a 1d solid with the assumption the electron can sit only on the atoms of the solid.
-
-The solid is assumed to consist of `N` sites with a seperation of `a` between them.
-
-## Refs.
-
-- https://www.damtp.cam.ac.uk/user/tong/aqm/aqmtwo.pdf
-
--/
-
-TODO "BBZAB" "Prove results related to the one-dimensional tight binding chain.
-    This is related to the following issue/feature-request:
-    https://github.com/HEPLean/PhysLean/issues/523 "
diff --git a/PhysLean/StringTheory/FTheory/SU5U1/Charges.lean b/PhysLean/StringTheory/FTheory/SU5U1/Charges.lean
index 343a9e3c2..19e684544 100644
--- a/PhysLean/StringTheory/FTheory/SU5U1/Charges.lean
+++ b/PhysLean/StringTheory/FTheory/SU5U1/Charges.lean
@@ -5,8 +5,6 @@ Authors: Joseph Tooby-Smith
 -/
 import Mathlib.Data.Finset.Insert
 import PhysLean.Meta.TODO.Basic
-import Mathlib.Data.Fintype.Defs
-import Mathlib.Data.Fintype.Sets
 /-!
 
 # Allowed charges
@@ -57,11 +55,6 @@ def allowedBarFiveCharges : CodimensionOneConfig → Finset ℤ
   | nearestNeighbor => {-14, -9, -4, 1, 6, 11}
   | nextToNearestNeighbor => {-13, -8, -3, 2, 7, 12}
 
-instance : (I : CodimensionOneConfig) → Fintype I.allowedBarFiveCharges
-  | same => inferInstance
-  | nearestNeighbor => inferInstance
-  | nextToNearestNeighbor => inferInstance
-
 /-- The allowed `U(1)`-charges of matter in the 10d representation of `SU(5)`
   given a `CodimensionOneConfig`. -/
 def allowedTenCharges : CodimensionOneConfig → Finset ℤ
@@ -69,11 +62,6 @@ def allowedTenCharges : CodimensionOneConfig → Finset ℤ
   | nearestNeighbor => {-12, -7, -2, 3, 8, 13}
   | nextToNearestNeighbor => {-9, -4, 1, 6, 11}
 
-instance : (I : CodimensionOneConfig) → Fintype I.allowedTenCharges
-  | same => inferInstance
-  | nearestNeighbor => inferInstance
-  | nextToNearestNeighbor => inferInstance
-
 end CodimensionOneConfig
 end SU5U1
 
diff --git a/PhysLean/StringTheory/FTheory/SU5U1/Examples.lean b/PhysLean/StringTheory/FTheory/SU5U1/Examples.lean
index 18ea0460f..393208739 100644
--- a/PhysLean/StringTheory/FTheory/SU5U1/Examples.lean
+++ b/PhysLean/StringTheory/FTheory/SU5U1/Examples.lean
@@ -54,10 +54,7 @@ def mkOneTenFourFiveBar (I : CodimensionOneConfig) (M : ChiralityFlux) (N : Hype
     (h5 : ∀ a ∈ ({(M, N, ⟨q5₁, hq5₁⟩), (3 - M, - N, ⟨q5₂, hq5₂⟩)} :
       Multiset (QuantaBarFive I)), a.M = 0 → a.N ≠ 0 := by decide)
     (h10 :  ∀ a ∈ ({(3, 0, ⟨q10, hq10⟩)} :
-      Multiset (QuantaTen I)), a.M = 0 → a.N ≠ 0 := by decide)
-    (hd5 : DistinctChargedBarFive (I := I) {(M, N, ⟨q5₁, hq5₁⟩), (3 - M, - N, ⟨q5₂, hq5₂⟩)}
-      ⟨-qHu, hqHu⟩ ⟨qHd, hqHd⟩ := by decide)
-    (hd10 : DistinctChargedTen (I := I) {(3, 0, ⟨q10, hq10⟩)} := by decide) :
+      Multiset (QuantaTen I)), a.M = 0 → a.N ≠ 0 := by decide) :
     MatterContent I where
   quantaTen := {(3, 0, ⟨q10, hq10⟩)}
   qHu := ⟨- qHu, hqHu⟩
@@ -66,16 +63,12 @@ def mkOneTenFourFiveBar (I : CodimensionOneConfig) (M : ChiralityFlux) (N : Hype
     (3 - M, - N, ⟨q5₂, hq5₂⟩)}
   chirality_charge_not_both_zero_bar_five_matter := h5
   chirality_charge_not_both_zero_ten := h10
-  distinctly_charged_quantaBarFiveMatter := hd5
-  distinctly_charged_quantaTen := hd10
 
 /-- An example of matter content with one 10d representation and 4 5-bar representations.
   This corresponds to example I.1.4.a in table 1 of arXiv:1507.05961. -/
 def caseI14a : MatterContent .same :=
   mkOneTenFourFiveBar .same 1 2 (-1) 2 2 (-1) 1
 
--- #eval println! caseI14a.phenoCharges
-
 /-- An example of matter content with one 10d representation and 4 5-bar representations.
   This corresponds to example I.1.4.b in table 1 of arXiv:1507.05961. -/
 def caseI14b : MatterContent .same :=
@@ -123,8 +116,6 @@ def caseI24a : MatterContent .same where
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
-  distinctly_charged_quantaBarFiveMatter := by decide
-  distinctly_charged_quantaTen := by decide
 
 /-- An example of matter content with two 10d representation and 4 5-bar representations.
   This corresponds to one of the two versions of the I.2.4.a in table 8 of arXiv:1507.05961. -/
@@ -137,8 +128,6 @@ def caseI24a' : MatterContent .same where
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
-  distinctly_charged_quantaBarFiveMatter := by decide
-  distinctly_charged_quantaTen := by decide
 
 /-- An example of matter content with two 10d representation and 4 5-bar representations.
   This corresponds to one of the four versions of I.2.4.b in table 8 of arXiv:1507.05961. -/
@@ -151,8 +140,6 @@ def caseI24b : MatterContent .same where
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
-  distinctly_charged_quantaBarFiveMatter := by decide
-  distinctly_charged_quantaTen := by decide
 
 /-- An example of matter content with two 10d representation and 4 5-bar representations.
   This corresponds to one of the four versions of I.2.4.b in table 8 of arXiv:1507.05961. -/
@@ -165,8 +152,6 @@ def caseI24b' : MatterContent .same where
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
-  distinctly_charged_quantaBarFiveMatter := by decide
-  distinctly_charged_quantaTen := by decide
 
 /-- An example of matter content with two 10d representation and 4 5-bar representations.
   This corresponds to one of the four versions of I.2.4.b in table 8 of arXiv:1507.05961. -/
@@ -179,8 +164,6 @@ def caseI24b'' : MatterContent .same where
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
-  distinctly_charged_quantaBarFiveMatter := by decide
-  distinctly_charged_quantaTen := by decide
 
 /-- An example of matter content with two 10d representation and 4 5-bar representations.
   This corresponds to one of the four versions of I.2.4.b in table 8 of arXiv:1507.05961. -/
@@ -193,8 +176,6 @@ def caseI24b''' : MatterContent .same where
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
-  distinctly_charged_quantaBarFiveMatter := by decide
-  distinctly_charged_quantaTen := by decide
 
 /-!
 
@@ -209,13 +190,12 @@ def caseI34a : MatterContent .same where
   quantaTen := {(1, 0, ⟨-3, by decide⟩), (1, 0, ⟨-2, by decide⟩), (1, 0, ⟨-1, by decide⟩)}
   qHu := ⟨-2, by decide⟩
   qHd := ⟨1, by decide⟩
-  quantaBarFiveMatter := {(0, 3, ⟨-1, by decide⟩), (3, -3, ⟨0, by decide⟩)}
+  quantaBarFiveMatter := {
+    (0, 3, ⟨-1, by decide⟩), (3, -3, ⟨0, by decide⟩)}
   chirality_charge_not_both_zero_bar_five_matter := by
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
-  distinctly_charged_quantaBarFiveMatter := by decide
-  distinctly_charged_quantaTen := by decide
 
 /-- The finite set of all examples of MatterContent currently defined in PhysLean. -/
 def allCases : Finset (Σ I, MatterContent I) :=
diff --git a/PhysLean/StringTheory/FTheory/SU5U1/Matter.lean b/PhysLean/StringTheory/FTheory/SU5U1/Matter.lean
index cd75425bd..471307565 100644
--- a/PhysLean/StringTheory/FTheory/SU5U1/Matter.lean
+++ b/PhysLean/StringTheory/FTheory/SU5U1/Matter.lean
@@ -7,8 +7,6 @@ import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
 import Mathlib.Algebra.Group.Int.Defs
 import Mathlib.Algebra.Order.Group.Unbundled.Abs
 import PhysLean.StringTheory.FTheory.SU5U1.Charges
-import Mathlib.Data.Finset.Card
-import Mathlib.Data.Finset.Powerset
 /-!
 
 # Matter
@@ -68,33 +66,6 @@ abbrev QuantaTen.N {I : CodimensionOneConfig} (a : QuantaTen I) : HyperChargeFlu
 abbrev QuantaTen.q {I : CodimensionOneConfig} (a : QuantaTen I) :
     I.allowedTenCharges := a.2.2
 
-/-- The proposition on `Multiset (QuantaBarFive I)`,
-  and two `I.allowedBarFiveCharges` denoted `qHu` and `qHd` which is true
-  if none of the (underlying) charges are equal. -/
-def DistinctChargedBarFive {I : CodimensionOneConfig}
-    (quantaBarFiveMatter : Multiset (QuantaBarFive I))
-    (qHu : I.allowedBarFiveCharges) (qHd : I.allowedBarFiveCharges) : Prop :=
-  (quantaBarFiveMatter.map QuantaBarFive.q).toFinset.card =
-      (quantaBarFiveMatter.map QuantaBarFive.q).card
-    ∧ qHu ∉ (quantaBarFiveMatter.map QuantaBarFive.q)
-    ∧ qHd ∉ (quantaBarFiveMatter.map QuantaBarFive.q)
-    ∧ qHu ≠ qHd
-
-instance {I : CodimensionOneConfig}
-    (quantaBarFiveMatter : Multiset (QuantaBarFive I))
-    (qHu : I.allowedBarFiveCharges) (qHd : I.allowedBarFiveCharges) :
-    Decidable (DistinctChargedBarFive quantaBarFiveMatter qHu qHd) := instDecidableAnd
-
-/-- The proposition on a `Multiset (QuantaTen I)` which is true if non of the underlying
-  charges are equal. -/
-def DistinctChargedTen {I : CodimensionOneConfig}
-    (quantaTen : Multiset (QuantaTen I)) : Prop :=
-  (quantaTen.map QuantaTen.q).toFinset.card = (quantaTen.map QuantaTen.q).card
-
-instance {I : CodimensionOneConfig}
-    (quantaTen : Multiset (QuantaTen I)) :
-    Decidable (DistinctChargedTen quantaTen) := decEq _ _
-
 /-- The matter content, assumed to sit in the 5-bar or 10d representation of
   `SU(5)`. -/
 @[ext]
@@ -112,10 +83,6 @@ structure MatterContent (I : CodimensionOneConfig) where
     ∀ a ∈ quantaBarFiveMatter, (a.M = 0 → a.N ≠ 0)
   /-- There is no matter in the 10d representation with zero `Chirality` and `HyperChargeFlux`. -/
   chirality_charge_not_both_zero_ten : ∀ a ∈ quantaTen, (a.M = 0 → a.N ≠ 0)
-  /-- All 5-bar representations carry distinct charges. -/
-  distinctly_charged_quantaBarFiveMatter : DistinctChargedBarFive quantaBarFiveMatter qHu qHd
-  /-- All 10d representations carry distinct charges. -/
-  distinctly_charged_quantaTen : DistinctChargedTen quantaTen
 
 namespace MatterContent
 
@@ -167,39 +134,6 @@ lemma quantaBarFive_chiralityFlux_two_le_filter_zero_card :
   funext x
   exact Lean.Grind.eq_congr' rfl rfl
 
-lemma quantaBarFiveMatter_map_q_noDup :
-    (𝓜.quantaBarFiveMatter.map (QuantaBarFive.q)).Nodup :=
-  Multiset.dedup_card_eq_card_iff_nodup.mp 𝓜.distinctly_charged_quantaBarFiveMatter.1
-
-lemma quantaBarFive_map_q_noDup : (𝓜.quantaBarFive.map (QuantaBarFive.q)).Nodup := by
-  simp only [quantaBarFive, Int.reduceNeg, Multiset.map_cons, Multiset.nodup_cons,
-    Multiset.mem_cons, Multiset.mem_map, Prod.exists, exists_eq_right, not_or, not_exists,
-    𝓜.quantaBarFiveMatter_map_q_noDup, and_true]
-  have h1 := 𝓜.distinctly_charged_quantaBarFiveMatter
-  simp_all only [DistinctChargedBarFive, QuantaBarFive.q, Multiset.card_map, Multiset.mem_map,
-    Prod.exists, exists_eq_right, not_exists, ne_eq, not_false_eq_true, implies_true, and_true]
-  exact fun a => h1.2.2.2 a.symm
-
-set_option maxRecDepth 1000 in
-lemma quantaBarFive_map_q_card_le_seven :
-    (𝓜.quantaBarFive.map (QuantaBarFive.q)).card ≤ 7 := by
-  rw [← Multiset.dedup_card_eq_card_iff_nodup.mpr 𝓜.quantaBarFive_map_q_noDup]
-  have h1 :  (Multiset.map QuantaBarFive.q 𝓜.quantaBarFive).toFinset ∈
-      Finset.powerset (Finset.univ (α := I.allowedBarFiveCharges)) := by
-    rw [Finset.mem_powerset]
-    exact Finset.subset_univ _
-  change (Multiset.map QuantaBarFive.q 𝓜.quantaBarFive).toFinset.card ≤ 7
-  generalize (Multiset.map QuantaBarFive.q 𝓜.quantaBarFive).toFinset = S at *
-  revert S
-  match I with
-  | CodimensionOneConfig.same => decide
-  | CodimensionOneConfig.nearestNeighbor => decide
-  | CodimensionOneConfig.nextToNearestNeighbor => decide
-
-lemma quantaBarFive_card_le_seven : 𝓜.quantaBarFive.card ≤ 7 := by
-  apply le_of_eq_of_le _ 𝓜.quantaBarFive_map_q_card_le_seven
-  simp
-
 /-!
 
 ## Gauge anomalies
diff --git a/PhysLean/StringTheory/FTheory/SU5U1/NoExotics/FiveBar.lean b/PhysLean/StringTheory/FTheory/SU5U1/NoExotics/FiveBar.lean
index bab38326f..673f622da 100644
--- a/PhysLean/StringTheory/FTheory/SU5U1/NoExotics/FiveBar.lean
+++ b/PhysLean/StringTheory/FTheory/SU5U1/NoExotics/FiveBar.lean
@@ -249,9 +249,7 @@ lemma quantaBarFive_zero_chiralityFlux_mem (h3L : 𝓜.ThreeLeptonDoublets) :
   · rw [hn 5 hr]
     simp
 
-/-- The number of 5-bar representations is less than or equal to eight.
-  Note the existences of `quantaBarFive_card_le_seven`. The proof of this weaker result
-  does not rely on the assumptions of no-duplicate charges. -/
+/-- The number of 5-bar representations is less than or equal to eight. -/
 lemma quantaBarFive_card_le_eight (h3 : 𝓜.ThreeChiralFamiles) (h3L : 𝓜.ThreeLeptonDoublets) :
     𝓜.quantaBarFive.card ≤ 8 := by
   have h1 : 𝓜.quantaBarFive.card =
@@ -276,15 +274,11 @@ with zero to five chirality fluxes equal to zero.
 -/
 lemma quantaBarFive_chiralityFlux_mem (h3 : 𝓜.ThreeChiralFamiles) (h3L : 𝓜.ThreeLeptonDoublets) :
     𝓜.quantaBarFive.map QuantaBarFive.M ∈
-    ({{1, 1, 1, 0, 0, 0, 0}, {1, 1, 1, 0, 0, 0}, {1, 1, 1, 0, 0},
+    ({{1, 1, 1, 0, 0, 0, 0, 0}, {1, 1, 1, 0, 0, 0, 0}, {1, 1, 1, 0, 0, 0}, {1, 1, 1, 0, 0},
       {1, 2, 0, 0, 0, 0, 0}, {1, 2, 0, 0, 0, 0}, {1, 2, 0, 0, 0}, {1, 2, 0, 0},
       {3, 0, 0, 0, 0, 0}, {3, 0, 0, 0, 0}, {3, 0, 0, 0}, {3, 0, 0}} :
       Finset (Multiset ChiralityFlux)) := by
-  have hcard : (𝓜.quantaBarFive.map QuantaBarFive.M).card ≤ 7 := by
-    rw [Multiset.card_map]
-    exact 𝓜.quantaBarFive_card_le_seven
-  rw [← (Multiset.filter_add_not (fun x => x = 0)
-    (Multiset.map QuantaBarFive.M 𝓜.quantaBarFive))] at hcard ⊢
+  rw [← (Multiset.filter_add_not (fun x => x = 0) (Multiset.map QuantaBarFive.M 𝓜.quantaBarFive))]
   have hz := quantaBarFive_zero_chiralityFlux_mem h3L
   have h0 := quantaBarFive_chiralityFlux_filter_non_zero_mem h3
   simp only [Finset.mem_insert, Finset.mem_singleton] at hz
@@ -301,9 +295,6 @@ lemma quantaBarFive_chiralityFlux_mem (h3 : 𝓜.ThreeChiralFamiles) (h3L : 𝓜
     rw [Multiset.add_comm, Multiset.singleton_add]
     simp only [Multiset.cons_zero, Finset.mem_insert, Finset.mem_singleton,
       Multiset.empty_eq_zero, true_or, or_true]
-  -- The case of {1, 1, 1, 0, 0, 0, 0, 0}
-  rw [hz, h0] at hcard
-  simp at hcard
 
 /-- The number of 5-bar representations (including the Higges) is greater then or equal to three. -/
 lemma quantaBarFive_three_le_card (h3 : 𝓜.ThreeChiralFamiles) (h3L : 𝓜.ThreeLeptonDoublets) :
diff --git a/PhysLean/StringTheory/FTheory/SU5U1/PhenoConstraints/Basic.lean b/PhysLean/StringTheory/FTheory/SU5U1/PhenoConstraints/Basic.lean
index eda81d259..e9813d4cf 100644
--- a/PhysLean/StringTheory/FTheory/SU5U1/PhenoConstraints/Basic.lean
+++ b/PhysLean/StringTheory/FTheory/SU5U1/PhenoConstraints/Basic.lean
@@ -43,93 +43,12 @@ In what follows we constrain via `U(1)` charges
 namespace FTheory
 
 namespace SU5U1
-
-variable {I : CodimensionOneConfig}
-
-/-- The overall charge of the term `μ 5Hu 5̄Hd` -/
-def chargeMuTerm (qHu qHd : I.allowedBarFiveCharges) : ℤ := - qHu.1 + qHd.1
-
-/-- The charges of the term `W¹ᵢⱼₖₗ 10ⁱ 10ʲ 10ᵏ 5̄Mˡ`. -/
-def chargeW1Term (q5 : Multiset I.allowedBarFiveCharges) (q10 : Multiset I.allowedTenCharges) :
-    Multiset ℤ :=
-  (Multiset.product q10 (Multiset.product q10 (Multiset.product q10 q5))).map
-  (fun x => x.1 + x.2.1 + x.2.2.1 + x.2.2.2)
-
-/-- The charges of the term `𝛽ᵢ 5̄Mⁱ5Hu`. -/
-def chargeBetaTerm (q5bar : Multiset I.allowedBarFiveCharges) (qHu : I.allowedBarFiveCharges) :
-    Multiset ℤ := q5bar.map (fun x => x.1 + (- qHu.1))
-
-/-- The charges of the term `𝜆ᵢⱼₖ 5̄Mⁱ 5̄Mʲ 10ᵏ`. -/
-def chargeLambdaTerm (q5bar : Multiset I.allowedBarFiveCharges)
-    (q10 : Multiset I.allowedTenCharges) : Multiset ℤ :=
-  (Multiset.product q5bar (Multiset.product q5bar q10)).map
-  (fun x => x.1 + x.2.1 + x.2.2.1)
-
-/-- The charges of the term `K¹ᵢⱼₖ  10ⁱ 10ʲ 5Mᵏ`. -/
-def chargeK1Term (q5bar : Multiset I.allowedBarFiveCharges)
-    (q10 : Multiset I.allowedTenCharges) : Multiset ℤ :=
-  (Multiset.product q10 (Multiset.product q10 q5bar)).map
-  (fun x => x.1 + x.2.1 + (- x.2.2.1))
-
-/-- The charges of the term `W⁴ᵢ 5̄Mⁱ 5̄Hd 5Hu 5Hu`. -/
-def chargeW4Term (q5bar : Multiset I.allowedBarFiveCharges)
-    (qHd : I.allowedBarFiveCharges) (qHu : I.allowedBarFiveCharges) : Multiset ℤ :=
-  q5bar.map (fun x => x.1 + qHd.1 + (- qHu.1) + (- qHu.1))
-
-/-- The charges of the term `K²ᵢ 5̄Hu 5̄Hd 10ⁱ` -/
-def chargeK2Term (q10 : Multiset I.allowedTenCharges)
-    (qHu : I.allowedBarFiveCharges) (qHd : I.allowedBarFiveCharges) :
-    Multiset ℤ :=
-  q10.map (fun x => qHu.1 + qHd.1 + x.1)
-
-/-- The charges of the term `W²ᵢⱼₖ 10ⁱ 10ʲ 10ᵏ 5̄Hd`. -/
-def chargeW2Term (q10 : Multiset I.allowedTenCharges)
-    (qHd : I.allowedBarFiveCharges) : Multiset ℤ :=
-  (Multiset.product q10 (Multiset.product q10 q10)).map
-  (fun x => x.1 + x.2.1 + x.2.2.1 + qHd.1)
-
-/-- The charges associated with the term `λᵗᵢⱼ 10ⁱ 10ʲ 5Hu`-/
-def chargeYukawaTop (q10 : Multiset I.allowedTenCharges)
-    (qHu : I.allowedBarFiveCharges) : Multiset ℤ :=
-  ((Multiset.product q10 q10)).map (fun x => x.1 + x.2.1 + (- qHu.1))
-
-/-- The charges associated with the term `λᵇᵢⱼ 10ⁱ 5̄Mʲ 5̄Hd``. -/
-def chargeYukawaBottom (q5bar : Multiset I.allowedBarFiveCharges)
-    (q10 : Multiset I.allowedTenCharges) (qHd : I.allowedBarFiveCharges) : Multiset ℤ :=
-  (Multiset.product q10 q5bar).map (fun x => x.1 + x.2.1 + qHd.1)
-
 namespace MatterContent
 variable {I : CodimensionOneConfig} (𝓜 : MatterContent I)
 
-/-- A string containing the U(1)-charges associated with interaction terms. -/
-def phenoCharges : String :=
-  s!"
-Charges associated with terms:
-μ-term (μ 5Hu 5̄Hd): {chargeMuTerm 𝓜.qHu 𝓜.qHd}
-W¹-term (W¹ᵢⱼₖₗ 10ⁱ 10ʲ 10ᵏ 5̄Mˡ): {(chargeW1Term (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
-  (𝓜.quantaTen.map QuantaTen.q)).sort (LE.le) }
-𝛽-term (𝛽ᵢ 5̄Mⁱ5Hu): {(chargeBetaTerm (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
-  𝓜.qHu).sort LE.le}
-𝜆-term (𝜆ᵢⱼₖ 5̄Mⁱ 5̄Mʲ 10ᵏ): {(chargeLambdaTerm (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
-  (𝓜.quantaTen.map QuantaTen.q)).sort LE.le}
-K¹-term (K¹ᵢⱼₖ  10ⁱ 10ʲ 5Mᵏ): {(chargeK1Term (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
-  (𝓜.quantaTen.map QuantaTen.q)).sort LE.le}
-W⁴-term (W⁴ᵢ 5̄Mⁱ 5̄Hd 5Hu 5Hu): {(chargeW4Term (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
-  𝓜.qHd 𝓜.qHu).sort LE.le}
-K²-term (K²ᵢ 5̄Hu 5̄Hd 10ⁱ): {(chargeK2Term (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHu
-  𝓜.qHd).sort LE.le}
-...
-W²-term (W²ᵢⱼₖ 10ⁱ 10ʲ 10ᵏ 5̄Hd): {(chargeW2Term (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHd).sort LE.le}
-...
-Top-Yukawa (λᵗᵢⱼ 10ⁱ 10ʲ 5Hu): {(chargeYukawaTop (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHu).sort LE.le}
-Bottom-Yukawa (λᵇᵢⱼ 10ⁱ 5̄Mʲ 5̄Hd): {(chargeYukawaBottom (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
-  (𝓜.quantaTen.map QuantaTen.q)
-   𝓜.qHd).sort LE.le}
-"
-
 /-- A proposition which is true when the `μ`-term (`5Hu 5̄Hd`) does not obey the additional
   `U(1)` symmetry in the model, and is therefore constrained. -/
-def MuTermU1Constrained : Prop := chargeMuTerm 𝓜.qHu 𝓜.qHd ≠ 0
+def MuTermU1Constrained : Prop := - 𝓜.qHu.1 + 𝓜.qHd.1 ≠ 0
 
 instance : Decidable 𝓜.MuTermU1Constrained := instDecidableNot
 
@@ -146,19 +65,20 @@ instance : Decidable 𝓜.MuTermU1Constrained := instDecidableNot
 -/
 def RParityU1Constrained : Prop :=
   --`𝛽ᵢ 5̄Mⁱ5Hu`
-  0 ∉ chargeBetaTerm (𝓜.quantaBarFiveMatter.map QuantaBarFive.q) 𝓜.qHu
+  (∀ fi ∈ 𝓜.quantaBarFiveMatter, fi.q.1 + (- 𝓜.qHu.1) ≠ 0)
   -- `𝜆ᵢⱼₖ 5̄Mⁱ 5̄Mʲ 10ᵏ`
-  ∧ 0 ∉ chargeLambdaTerm (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
-    (𝓜.quantaTen.map QuantaTen.q)
+  ∧ (∀ fi ∈ 𝓜.quantaBarFiveMatter, ∀ fj ∈ 𝓜.quantaBarFiveMatter, ∀ tk ∈ 𝓜.quantaTen,
+    fi.q.1 + fj.q.1 + tk.q.1 ≠ 0)
   -- `W²ᵢⱼₖ 10ⁱ 10ʲ 10ᵏ 5̄Hd`
-  ∧ 0 ∉ chargeW2Term (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHd
+  ∧ (∀ ti ∈ 𝓜.quantaTen, ∀ tj ∈ 𝓜.quantaTen, ∀ tk ∈ 𝓜.quantaTen,
+    ti.q.1 + tj.q.1 + tk.q.1 + 𝓜.qHd.1 ≠ 0)
   -- `W⁴ᵢ 5̄Mⁱ 5̄Hd 5Hu 5Hu`
-  ∧ 0 ∉ chargeW4Term (𝓜.quantaBarFiveMatter.map QuantaBarFive.q) 𝓜.qHd 𝓜.qHu
+  ∧ (∀ fi ∈ 𝓜.quantaBarFiveMatter, fi.q.1 + 𝓜.qHd.1 + (- 𝓜.qHu.1) + (- 𝓜.qHu.1) ≠ 0)
   -- `K¹ᵢⱼₖ  10ⁱ 10ʲ 5Mᵏ`
-  ∧ 0 ∉ chargeK1Term (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
-    (𝓜.quantaTen.map QuantaTen.q)
+  ∧ (∀ ti ∈ 𝓜.quantaTen, ∀ tj ∈ 𝓜.quantaTen, ∀ fk ∈ 𝓜.quantaBarFiveMatter,
+    ti.q.1 + tj.q.1 + (- fk.q.1) ≠ 0)
   -- `K²ᵢ 5̄Hu 5̄Hd 10ⁱ`
-  ∧ 0 ∉ chargeK2Term (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHu 𝓜.qHd
+  ∧ (∀ ti ∈ 𝓜.quantaTen, 𝓜.qHu.1 + 𝓜.qHd.1 + ti.q.1 ≠ 0)
 
 instance : Decidable 𝓜.RParityU1Constrained := instDecidableAnd
 
@@ -173,37 +93,44 @@ instance : Decidable 𝓜.RParityU1Constrained := instDecidableAnd
 -/
 def ProtonDecayU1Constrained : Prop :=
   -- `W¹ᵢⱼₖₗ 10ⁱ 10ʲ 10ᵏ 5̄Mˡ`
-  0 ∉ chargeW1Term (𝓜.quantaBarFiveMatter.map QuantaBarFive.q) (𝓜.quantaTen.map QuantaTen.q)
+  (∀ ti ∈ 𝓜.quantaTen, ∀ tj ∈ 𝓜.quantaTen, ∀ tk ∈ 𝓜.quantaTen, ∀ fl ∈ 𝓜.quantaBarFiveMatter,
+    ti.q.1 + tj.q.1 + tk.q.1 + fl.q.1 ≠ 0)
   -- `𝜆ᵢⱼₖ 5̄Mⁱ 5̄Mʲ 10ᵏ`
-  ∧ 0 ∉ chargeLambdaTerm (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
-    (𝓜.quantaTen.map QuantaTen.q)
+  ∧ (∀ fi ∈ 𝓜.quantaBarFiveMatter, ∀ fj ∈ 𝓜.quantaBarFiveMatter, ∀ tk ∈ 𝓜.quantaTen,
+    fi.q.1 + fj.q.1 + tk.q.1 ≠ 0)
   -- `W²ᵢⱼₖ 10ⁱ 10ʲ 10ᵏ 5̄Hd`
-  ∧ 0 ∉ chargeW2Term (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHd
+  ∧ (∀ ti ∈ 𝓜.quantaTen, ∀ tj ∈ 𝓜.quantaTen, ∀ tk ∈ 𝓜.quantaTen,
+    ti.q.1 + tj.q.1 + tk.q.1 + 𝓜.qHd.1 ≠ 0)
   -- `K¹ᵢⱼₖ  10ⁱ 10ʲ 5Mᵏ`
-  ∧ 0 ∉ chargeK1Term (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
-    (𝓜.quantaTen.map QuantaTen.q)
+  ∧ (∀ ti ∈ 𝓜.quantaTen, ∀ tj ∈ 𝓜.quantaTen, ∀ fk ∈ 𝓜.quantaBarFiveMatter,
+    ti.q.1 + tj.q.1 + (- fk.q.1) ≠ 0)
 
 instance : Decidable 𝓜.ProtonDecayU1Constrained := instDecidableAnd
 
 /-- The condition on the matter content for there to exist at least one copy of the coupling
 - `λᵗᵢⱼ 10ⁱ 10ʲ 5Hu`
 -/
-def HasATopYukawa  (𝓜 : MatterContent I) : Prop :=
-  0 ∈ chargeYukawaTop (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHu
+def HasATopYukawa  (𝓜 : MatterContent I) : Prop := ∃ ti ∈ 𝓜.quantaTen,  ∃ tj ∈ 𝓜.quantaTen,
+  ti.q.1 + tj.q.1 + (- 𝓜.qHu.1)  = 0
 
 instance : Decidable 𝓜.HasATopYukawa :=
-  Multiset.decidableMem 0 (chargeYukawaTop (Multiset.map QuantaTen.q 𝓜.quantaTen) 𝓜.qHu)
-
+  haveI : DecidablePred fun (ti : QuantaTen I) =>
+      ∃ tj ∈ 𝓜.quantaTen, ti.q.1 + ↑tj.q + -↑𝓜.qHu = 0 := fun _ =>
+        Multiset.decidableExistsMultiset
+  Multiset.decidableExistsMultiset
 
 /-- The condition on the matter content for there to exist at least one copy of the coupling
 - `λᵇᵢⱼ 10ⁱ 5̄Mʲ 5̄Hd`
 -/
-def HasABottomYukawa (𝓜 : MatterContent I) : Prop :=
-  0 ∈ chargeYukawaBottom (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
-    (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHu
+def HasABottomYukawa (𝓜 : MatterContent I) : Prop := ∃ ti ∈ 𝓜.quantaTen,
+  ∃ fj ∈ 𝓜.quantaBarFiveMatter,
+  ti.q.1 + fj.q.1 + 𝓜.qHd.1 = 0
 
 instance : Decidable 𝓜.HasABottomYukawa :=
-  Multiset.decidableMem _ _
+  haveI : DecidablePred fun (ti : QuantaTen I) =>
+      ∃ fj ∈ 𝓜.quantaBarFiveMatter, ti.q.1 + fj.q.1 + 𝓜.qHd.1 = 0 := fun _ =>
+        Multiset.decidableExistsMultiset
+  Multiset.decidableExistsMultiset
 
 end MatterContent
 end SU5U1

From 425300643805c6b7c15ff7d84321cabe2a342f0b Mon Sep 17 00:00:00 2001
From: jstoobysmith <72603918+jstoobysmith@users.noreply.github.com>
Date: Fri, 2 May 2025 07:21:01 +0000
Subject: [PATCH 20/32] chore: Fix file overwrites

---
 .../HarmonicOscillator/Completeness.lean      |  11 +-
 .../StringTheory/FTheory/SU5U1/Charges.lean   |  12 ++
 .../StringTheory/FTheory/SU5U1/Examples.lean  |  21 ++-
 .../StringTheory/FTheory/SU5U1/Matter.lean    |  50 +++++++
 .../FTheory/SU5U1/NoExotics/FiveBar.lean      |  15 ++-
 .../FTheory/SU5U1/PhenoConstraints/Basic.lean | 126 +++++++++++++++---
 6 files changed, 203 insertions(+), 32 deletions(-)

diff --git a/PhysLean/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean b/PhysLean/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean
index cc3c777ee..57c9c2dad 100644
--- a/PhysLean/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean
+++ b/PhysLean/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean
@@ -125,14 +125,9 @@ lemma mul_power_integrable (f : ℝ → ℂ) (hf : MemHS f) (r : ℕ) :
     suffices h2 : IsUnit (↑((1/Q.ξ)^ r : ℂ)) by
       rw [IsUnit.integrable_smul_iff h2] at h1
       simpa using h1
-    simp only [isUnit_iff_ne_zero, ne_eq, pow_eq_zero_iff', Complex.ofReal_eq_zero]
-    intro h
-    cases h with
-    | intro h =>
-      have h' : Q.ξ = 0 := by
-        simp [one_div] at h
-      exact Q.ξ_pos.ne' h'
-
+    simp only [isUnit_iff_ne_zero, ne_eq, pow_eq_zero_iff', Complex.ofReal_eq_zero, not_and,
+      Decidable.not_not]
+    simp
   · simp only [ne_eq, Decidable.not_not] at hr
     subst hr
     simpa using Q.mul_physHermite_integrable f hf 0
diff --git a/PhysLean/StringTheory/FTheory/SU5U1/Charges.lean b/PhysLean/StringTheory/FTheory/SU5U1/Charges.lean
index 19e684544..343a9e3c2 100644
--- a/PhysLean/StringTheory/FTheory/SU5U1/Charges.lean
+++ b/PhysLean/StringTheory/FTheory/SU5U1/Charges.lean
@@ -5,6 +5,8 @@ Authors: Joseph Tooby-Smith
 -/
 import Mathlib.Data.Finset.Insert
 import PhysLean.Meta.TODO.Basic
+import Mathlib.Data.Fintype.Defs
+import Mathlib.Data.Fintype.Sets
 /-!
 
 # Allowed charges
@@ -55,6 +57,11 @@ def allowedBarFiveCharges : CodimensionOneConfig → Finset ℤ
   | nearestNeighbor => {-14, -9, -4, 1, 6, 11}
   | nextToNearestNeighbor => {-13, -8, -3, 2, 7, 12}
 
+instance : (I : CodimensionOneConfig) → Fintype I.allowedBarFiveCharges
+  | same => inferInstance
+  | nearestNeighbor => inferInstance
+  | nextToNearestNeighbor => inferInstance
+
 /-- The allowed `U(1)`-charges of matter in the 10d representation of `SU(5)`
   given a `CodimensionOneConfig`. -/
 def allowedTenCharges : CodimensionOneConfig → Finset ℤ
@@ -62,6 +69,11 @@ def allowedTenCharges : CodimensionOneConfig → Finset ℤ
   | nearestNeighbor => {-12, -7, -2, 3, 8, 13}
   | nextToNearestNeighbor => {-9, -4, 1, 6, 11}
 
+instance : (I : CodimensionOneConfig) → Fintype I.allowedTenCharges
+  | same => inferInstance
+  | nearestNeighbor => inferInstance
+  | nextToNearestNeighbor => inferInstance
+
 end CodimensionOneConfig
 end SU5U1
 
diff --git a/PhysLean/StringTheory/FTheory/SU5U1/Examples.lean b/PhysLean/StringTheory/FTheory/SU5U1/Examples.lean
index 4659c01e0..02804e6e7 100644
--- a/PhysLean/StringTheory/FTheory/SU5U1/Examples.lean
+++ b/PhysLean/StringTheory/FTheory/SU5U1/Examples.lean
@@ -66,12 +66,16 @@ def mkOneTenFourFiveBar (I : CodimensionOneConfig) (M : ChiralityFlux) (N : Hype
     (3 - M, - N, ⟨q5₂, hq5₂⟩)}
   chirality_charge_not_both_zero_bar_five_matter := h5
   chirality_charge_not_both_zero_ten := h10
+  distinctly_charged_quantaBarFiveMatter := hd5
+  distinctly_charged_quantaTen := hd10
 
 /-- An example of matter content with one 10d representation and 4 5-bar representations.
   This corresponds to example I.1.4.a in table 1 of arXiv:1507.05961. -/
 def caseI14a : MatterContent .same :=
   mkOneTenFourFiveBar .same 1 2 (-1) 2 2 (-1) 1
 
+-- #eval println! caseI14a.phenoCharges
+
 /-- An example of matter content with one 10d representation and 4 5-bar representations.
   This corresponds to example I.1.4.b in table 1 of arXiv:1507.05961. -/
 def caseI14b : MatterContent .same :=
@@ -119,6 +123,8 @@ def caseI24a : MatterContent .same where
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
+  distinctly_charged_quantaBarFiveMatter := by decide
+  distinctly_charged_quantaTen := by decide
 
 /-- An example of matter content with two 10d representation and 4 5-bar representations.
   This corresponds to one of the two versions of the I.2.4.a in table 8 of arXiv:1507.05961. -/
@@ -131,6 +137,8 @@ def caseI24a' : MatterContent .same where
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
+  distinctly_charged_quantaBarFiveMatter := by decide
+  distinctly_charged_quantaTen := by decide
 
 /-- An example of matter content with two 10d representation and 4 5-bar representations.
   This corresponds to one of the four versions of I.2.4.b in table 8 of arXiv:1507.05961. -/
@@ -143,6 +151,8 @@ def caseI24b : MatterContent .same where
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
+  distinctly_charged_quantaBarFiveMatter := by decide
+  distinctly_charged_quantaTen := by decide
 
 /-- An example of matter content with two 10d representation and 4 5-bar representations.
   This corresponds to one of the four versions of I.2.4.b in table 8 of arXiv:1507.05961. -/
@@ -155,6 +165,8 @@ def caseI24b' : MatterContent .same where
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
+  distinctly_charged_quantaBarFiveMatter := by decide
+  distinctly_charged_quantaTen := by decide
 
 /-- An example of matter content with two 10d representation and 4 5-bar representations.
   This corresponds to one of the four versions of I.2.4.b in table 8 of arXiv:1507.05961. -/
@@ -167,6 +179,8 @@ def caseI24b'' : MatterContent .same where
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
+  distinctly_charged_quantaBarFiveMatter := by decide
+  distinctly_charged_quantaTen := by decide
 
 /-- An example of matter content with two 10d representation and 4 5-bar representations.
   This corresponds to one of the four versions of I.2.4.b in table 8 of arXiv:1507.05961. -/
@@ -179,6 +193,8 @@ def caseI24b''' : MatterContent .same where
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
+  distinctly_charged_quantaBarFiveMatter := by decide
+  distinctly_charged_quantaTen := by decide
 
 /-!
 
@@ -192,12 +208,13 @@ def caseI34a : MatterContent .same where
   quantaTen := {(1, 0, ⟨-3, by decide⟩), (1, 0, ⟨-2, by decide⟩), (1, 0, ⟨-1, by decide⟩)}
   qHu := ⟨-2, by decide⟩
   qHd := ⟨1, by decide⟩
-  quantaBarFiveMatter := {
-    (0, 3, ⟨-1, by decide⟩), (3, -3, ⟨0, by decide⟩)}
+  quantaBarFiveMatter := {(0, 3, ⟨-1, by decide⟩), (3, -3, ⟨0, by decide⟩)}
   chirality_charge_not_both_zero_bar_five_matter := by
     simp [QuantaBarFive.N]
   chirality_charge_not_both_zero_ten := by
     simp [QuantaTen.N, QuantaTen.M]
+  distinctly_charged_quantaBarFiveMatter := by decide
+  distinctly_charged_quantaTen := by decide
 
 /-- The finite set of all examples of MatterContent currently defined in PhysLean. -/
 def allCases : Finset (Σ I, MatterContent I) :=
diff --git a/PhysLean/StringTheory/FTheory/SU5U1/Matter.lean b/PhysLean/StringTheory/FTheory/SU5U1/Matter.lean
index b32d95be6..be6addd63 100644
--- a/PhysLean/StringTheory/FTheory/SU5U1/Matter.lean
+++ b/PhysLean/StringTheory/FTheory/SU5U1/Matter.lean
@@ -7,6 +7,8 @@ import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
 import Mathlib.Algebra.Group.Int.Defs
 import Mathlib.Algebra.Order.Group.Unbundled.Abs
 import PhysLean.StringTheory.FTheory.SU5U1.Charges
+import Mathlib.Data.Finset.Card
+import Mathlib.Data.Finset.Powerset
 /-!
 
 # Matter
@@ -66,6 +68,33 @@ abbrev QuantaTen.N {I : CodimensionOneConfig} (a : QuantaTen I) : HyperChargeFlu
 abbrev QuantaTen.q {I : CodimensionOneConfig} (a : QuantaTen I) :
     I.allowedTenCharges := a.2.2
 
+/-- The proposition on `Multiset (QuantaBarFive I)`,
+  and two `I.allowedBarFiveCharges` denoted `qHu` and `qHd` which is true
+  if none of the (underlying) charges are equal. -/
+def DistinctChargedBarFive {I : CodimensionOneConfig}
+    (quantaBarFiveMatter : Multiset (QuantaBarFive I))
+    (qHu : I.allowedBarFiveCharges) (qHd : I.allowedBarFiveCharges) : Prop :=
+  (quantaBarFiveMatter.map QuantaBarFive.q).toFinset.card =
+      (quantaBarFiveMatter.map QuantaBarFive.q).card
+    ∧ qHu ∉ (quantaBarFiveMatter.map QuantaBarFive.q)
+    ∧ qHd ∉ (quantaBarFiveMatter.map QuantaBarFive.q)
+    ∧ qHu ≠ qHd
+
+instance {I : CodimensionOneConfig}
+    (quantaBarFiveMatter : Multiset (QuantaBarFive I))
+    (qHu : I.allowedBarFiveCharges) (qHd : I.allowedBarFiveCharges) :
+    Decidable (DistinctChargedBarFive quantaBarFiveMatter qHu qHd) := instDecidableAnd
+
+/-- The proposition on a `Multiset (QuantaTen I)` which is true if non of the underlying
+  charges are equal. -/
+def DistinctChargedTen {I : CodimensionOneConfig}
+    (quantaTen : Multiset (QuantaTen I)) : Prop :=
+  (quantaTen.map QuantaTen.q).toFinset.card = (quantaTen.map QuantaTen.q).card
+
+instance {I : CodimensionOneConfig}
+    (quantaTen : Multiset (QuantaTen I)) :
+    Decidable (DistinctChargedTen quantaTen) := decEq _ _
+
 /-- The matter content, assumed to sit in the 5-bar or 10d representation of
   `SU(5)`. -/
 @[ext]
@@ -83,6 +112,10 @@ structure MatterContent (I : CodimensionOneConfig) where
     ∀ a ∈ quantaBarFiveMatter, (a.M = 0 → a.N ≠ 0)
   /-- There is no matter in the 10d representation with zero `Chirality` and `HyperChargeFlux`. -/
   chirality_charge_not_both_zero_ten : ∀ a ∈ quantaTen, (a.M = 0 → a.N ≠ 0)
+  /-- All 5-bar representations carry distinct charges. -/
+  distinctly_charged_quantaBarFiveMatter : DistinctChargedBarFive quantaBarFiveMatter qHu qHd
+  /-- All 10d representations carry distinct charges. -/
+  distinctly_charged_quantaTen : DistinctChargedTen quantaTen
 
 namespace MatterContent
 
@@ -146,6 +179,23 @@ lemma quantaBarFiveMatter_map_q_eq_toFinset :
   conv_lhs => rw [← h1]
   rfl
 
+lemma quantaBarFiveMatter_map_q_mem_powerset :
+    (𝓜.quantaBarFiveMatter.map (QuantaBarFive.q)).toFinset ∈
+      Finset.powerset (Finset.univ (α := I.allowedBarFiveCharges)) := by
+  rw [Finset.mem_powerset]
+  exact Finset.subset_univ _
+
+lemma quantaBarFiveMatter_map_q_mem_powerset_filter_card {n : ℕ}
+    (hcard : 𝓜.quantaBarFiveMatter.card = n) :
+    (𝓜.quantaBarFiveMatter.map (QuantaBarFive.q)).toFinset ∈
+      (Finset.univ (α := I.allowedBarFiveCharges)).powerset.filter fun x => x.card = n := by
+  simp only [Finset.mem_filter, Finset.mem_powerset, Finset.subset_univ, true_and]
+  trans (𝓜.quantaBarFiveMatter.map (QuantaBarFive.q)).card
+  · rw [quantaBarFiveMatter_map_q_eq_toFinset]
+    simp only [Multiset.toFinset_val, Multiset.toFinset_dedup]
+    rfl
+  · simpa using hcard
+
 lemma quantaBarFive_map_q_noDup : (𝓜.quantaBarFive.map (QuantaBarFive.q)).Nodup := by
   simp only [quantaBarFive, Int.reduceNeg, Multiset.map_cons, Multiset.nodup_cons,
     Multiset.mem_cons, Multiset.mem_map, Prod.exists, exists_eq_right, not_or, not_exists,
diff --git a/PhysLean/StringTheory/FTheory/SU5U1/NoExotics/FiveBar.lean b/PhysLean/StringTheory/FTheory/SU5U1/NoExotics/FiveBar.lean
index 673f622da..bab38326f 100644
--- a/PhysLean/StringTheory/FTheory/SU5U1/NoExotics/FiveBar.lean
+++ b/PhysLean/StringTheory/FTheory/SU5U1/NoExotics/FiveBar.lean
@@ -249,7 +249,9 @@ lemma quantaBarFive_zero_chiralityFlux_mem (h3L : 𝓜.ThreeLeptonDoublets) :
   · rw [hn 5 hr]
     simp
 
-/-- The number of 5-bar representations is less than or equal to eight. -/
+/-- The number of 5-bar representations is less than or equal to eight.
+  Note the existences of `quantaBarFive_card_le_seven`. The proof of this weaker result
+  does not rely on the assumptions of no-duplicate charges. -/
 lemma quantaBarFive_card_le_eight (h3 : 𝓜.ThreeChiralFamiles) (h3L : 𝓜.ThreeLeptonDoublets) :
     𝓜.quantaBarFive.card ≤ 8 := by
   have h1 : 𝓜.quantaBarFive.card =
@@ -274,11 +276,15 @@ with zero to five chirality fluxes equal to zero.
 -/
 lemma quantaBarFive_chiralityFlux_mem (h3 : 𝓜.ThreeChiralFamiles) (h3L : 𝓜.ThreeLeptonDoublets) :
     𝓜.quantaBarFive.map QuantaBarFive.M ∈
-    ({{1, 1, 1, 0, 0, 0, 0, 0}, {1, 1, 1, 0, 0, 0, 0}, {1, 1, 1, 0, 0, 0}, {1, 1, 1, 0, 0},
+    ({{1, 1, 1, 0, 0, 0, 0}, {1, 1, 1, 0, 0, 0}, {1, 1, 1, 0, 0},
       {1, 2, 0, 0, 0, 0, 0}, {1, 2, 0, 0, 0, 0}, {1, 2, 0, 0, 0}, {1, 2, 0, 0},
       {3, 0, 0, 0, 0, 0}, {3, 0, 0, 0, 0}, {3, 0, 0, 0}, {3, 0, 0}} :
       Finset (Multiset ChiralityFlux)) := by
-  rw [← (Multiset.filter_add_not (fun x => x = 0) (Multiset.map QuantaBarFive.M 𝓜.quantaBarFive))]
+  have hcard : (𝓜.quantaBarFive.map QuantaBarFive.M).card ≤ 7 := by
+    rw [Multiset.card_map]
+    exact 𝓜.quantaBarFive_card_le_seven
+  rw [← (Multiset.filter_add_not (fun x => x = 0)
+    (Multiset.map QuantaBarFive.M 𝓜.quantaBarFive))] at hcard ⊢
   have hz := quantaBarFive_zero_chiralityFlux_mem h3L
   have h0 := quantaBarFive_chiralityFlux_filter_non_zero_mem h3
   simp only [Finset.mem_insert, Finset.mem_singleton] at hz
@@ -295,6 +301,9 @@ lemma quantaBarFive_chiralityFlux_mem (h3 : 𝓜.ThreeChiralFamiles) (h3L : 𝓜
     rw [Multiset.add_comm, Multiset.singleton_add]
     simp only [Multiset.cons_zero, Finset.mem_insert, Finset.mem_singleton,
       Multiset.empty_eq_zero, true_or, or_true]
+  -- The case of {1, 1, 1, 0, 0, 0, 0, 0}
+  rw [hz, h0] at hcard
+  simp at hcard
 
 /-- The number of 5-bar representations (including the Higges) is greater then or equal to three. -/
 lemma quantaBarFive_three_le_card (h3 : 𝓜.ThreeChiralFamiles) (h3L : 𝓜.ThreeLeptonDoublets) :
diff --git a/PhysLean/StringTheory/FTheory/SU5U1/PhenoConstraints/Basic.lean b/PhysLean/StringTheory/FTheory/SU5U1/PhenoConstraints/Basic.lean
index 7fc0181e5..547332ee0 100644
--- a/PhysLean/StringTheory/FTheory/SU5U1/PhenoConstraints/Basic.lean
+++ b/PhysLean/StringTheory/FTheory/SU5U1/PhenoConstraints/Basic.lean
@@ -74,6 +74,14 @@ lemma chargeW1Term_single_q10 (q5 : Multiset I.allowedBarFiveCharges)
     exact Multiset.singleton_subset.mpr ha
   exact fun a => h (h1 a)
 
+lemma chargeW1Term_subset_q10 (q5 : Multiset I.allowedBarFiveCharges)
+    (q10 : Multiset I.allowedTenCharges) (h : 0 ∉ chargeW1Term q5 q10)
+    (S : Multiset I.allowedTenCharges) (hS : S ⊆ q10) :
+    0 ∉ chargeW1Term q5 S := by
+  have h1 : chargeW1Term q5 S ⊆ chargeW1Term q5 q10 := by
+    apply chargeW1Term_subset_of_subset_ten
+    exact hS
+  exact fun a => h (h1 a)
 /-- The charges of the term `𝛽ᵢ 5̄Mⁱ5Hu`. -/
 def chargeBetaTerm (q5bar : Multiset I.allowedBarFiveCharges) (qHu : I.allowedBarFiveCharges) :
     Multiset ℤ := q5bar.map (fun x => x.1 + (- qHu.1))
@@ -104,6 +112,15 @@ lemma chargeLambdaTerm_single_q10 (q5 : Multiset I.allowedBarFiveCharges)
     exact Multiset.singleton_subset.mpr ha
   exact fun a => h (h1 a)
 
+lemma chargeLambdaTerm_subset_q10 (q5 : Multiset I.allowedBarFiveCharges)
+    (q10 : Multiset I.allowedTenCharges) (h : 0 ∉ chargeLambdaTerm q5 q10)
+    (S : Multiset I.allowedTenCharges) (hS : S ⊆ q10) :
+    0 ∉ chargeLambdaTerm q5 S := by
+  have h1 : chargeLambdaTerm q5 S ⊆ chargeLambdaTerm q5 q10 := by
+    apply chargeLambdaTerm_subset_of_subset_ten
+    exact hS
+  exact fun a => h (h1 a)
+
 /-- The charges of the term `K¹ᵢⱼₖ 10ⁱ 10ʲ 5Mᵏ`. -/
 def chargeK1Term (q5bar : Multiset I.allowedBarFiveCharges)
     (q10 : Multiset I.allowedTenCharges) : Multiset ℤ :=
@@ -130,6 +147,15 @@ lemma chargeK1Term_single_q10 (q5 : Multiset I.allowedBarFiveCharges)
     exact Multiset.singleton_subset.mpr ha
   exact fun a => h (h1 a)
 
+lemma chargeK1Term_subset_q10 (q5 : Multiset I.allowedBarFiveCharges)
+    (q10 : Multiset I.allowedTenCharges) (h : 0 ∉ chargeK1Term q5 q10)
+    (S : Multiset I.allowedTenCharges) (hS : S ⊆ q10) :
+    0 ∉ chargeK1Term q5 S := by
+  have h1 : chargeK1Term q5 S ⊆ chargeK1Term q5 q10 := by
+    apply chargeK1Term_subset_of_subset_ten
+    exact hS
+  exact fun a => h (h1 a)
+
 /-- The charges of the term `W⁴ᵢ 5̄Mⁱ 5̄Hd 5Hu 5Hu`. -/
 def chargeW4Term (q5bar : Multiset I.allowedBarFiveCharges)
     (qHd : I.allowedBarFiveCharges) (qHu : I.allowedBarFiveCharges) : Multiset ℤ :=
@@ -189,7 +215,7 @@ Bottom-Yukawa (λᵇᵢⱼ 10ⁱ 5̄Mʲ 5̄Hd) : {(chargeYukawaBottom
 
 /-- A proposition which is true when the `μ`-term (`5Hu 5̄Hd`) does not obey the additional
   `U(1)` symmetry in the model, and is therefore constrained. -/
-def MuTermU1Constrained : Prop := - 𝓜.qHu.1 + 𝓜.qHd.1 ≠ 0
+def MuTermU1Constrained : Prop := chargeMuTerm 𝓜.qHu 𝓜.qHd ≠ 0
 
 instance : Decidable 𝓜.MuTermU1Constrained := instDecidableNot
 
@@ -206,20 +232,19 @@ instance : Decidable 𝓜.MuTermU1Constrained := instDecidableNot
 -/
 def RParityU1Constrained : Prop :=
   --`𝛽ᵢ 5̄Mⁱ5Hu`
-  (∀ fi ∈ 𝓜.quantaBarFiveMatter, fi.q.1 + (- 𝓜.qHu.1) ≠ 0)
+  0 ∉ chargeBetaTerm (𝓜.quantaBarFiveMatter.map QuantaBarFive.q) 𝓜.qHu
   -- `𝜆ᵢⱼₖ 5̄Mⁱ 5̄Mʲ 10ᵏ`
-  ∧ (∀ fi ∈ 𝓜.quantaBarFiveMatter, ∀ fj ∈ 𝓜.quantaBarFiveMatter, ∀ tk ∈ 𝓜.quantaTen,
-    fi.q.1 + fj.q.1 + tk.q.1 ≠ 0)
+  ∧ 0 ∉ chargeLambdaTerm (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
+    (𝓜.quantaTen.map QuantaTen.q)
   -- `W²ᵢⱼₖ 10ⁱ 10ʲ 10ᵏ 5̄Hd`
-  ∧ (∀ ti ∈ 𝓜.quantaTen, ∀ tj ∈ 𝓜.quantaTen, ∀ tk ∈ 𝓜.quantaTen,
-    ti.q.1 + tj.q.1 + tk.q.1 + 𝓜.qHd.1 ≠ 0)
+  ∧ 0 ∉ chargeW2Term (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHd
   -- `W⁴ᵢ 5̄Mⁱ 5̄Hd 5Hu 5Hu`
   ∧ 0 ∉ chargeW4Term (𝓜.quantaBarFiveMatter.map QuantaBarFive.q) 𝓜.qHd 𝓜.qHu
   -- `K¹ᵢⱼₖ 10ⁱ 10ʲ 5Mᵏ`
   ∧ 0 ∉ chargeK1Term (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
     (𝓜.quantaTen.map QuantaTen.q)
   -- `K²ᵢ 5̄Hu 5̄Hd 10ⁱ`
-  ∧ (∀ ti ∈ 𝓜.quantaTen, 𝓜.qHu.1 + 𝓜.qHd.1 + ti.q.1 ≠ 0)
+  ∧ 0 ∉ chargeK2Term (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHu 𝓜.qHd
 
 instance : Decidable 𝓜.RParityU1Constrained := instDecidableAnd
 
@@ -234,11 +259,10 @@ instance : Decidable 𝓜.RParityU1Constrained := instDecidableAnd
 -/
 def ProtonDecayU1Constrained : Prop :=
   -- `W¹ᵢⱼₖₗ 10ⁱ 10ʲ 10ᵏ 5̄Mˡ`
-  (∀ ti ∈ 𝓜.quantaTen, ∀ tj ∈ 𝓜.quantaTen, ∀ tk ∈ 𝓜.quantaTen, ∀ fl ∈ 𝓜.quantaBarFiveMatter,
-    ti.q.1 + tj.q.1 + tk.q.1 + fl.q.1 ≠ 0)
+  0 ∉ chargeW1Term (𝓜.quantaBarFiveMatter.map QuantaBarFive.q) (𝓜.quantaTen.map QuantaTen.q)
   -- `𝜆ᵢⱼₖ 5̄Mⁱ 5̄Mʲ 10ᵏ`
-  ∧ (∀ fi ∈ 𝓜.quantaBarFiveMatter, ∀ fj ∈ 𝓜.quantaBarFiveMatter, ∀ tk ∈ 𝓜.quantaTen,
-    fi.q.1 + fj.q.1 + tk.q.1 ≠ 0)
+  ∧ 0 ∉ chargeLambdaTerm (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
+    (𝓜.quantaTen.map QuantaTen.q)
   -- `W²ᵢⱼₖ 10ⁱ 10ʲ 10ᵏ 5̄Hd`
   ∧ 0 ∉ chargeW2Term (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHd
   -- `K¹ᵢⱼₖ 10ⁱ 10ʲ 5Mᵏ`
@@ -250,7 +274,6 @@ instance : Decidable 𝓜.ProtonDecayU1Constrained := instDecidableAnd
 /-- The condition on the matter content for there to exist at least one copy of the coupling
 - `λᵗᵢⱼ 10ⁱ 10ʲ 5Hu`
 -/
-
 def HasATopYukawa (𝓜 : MatterContent I) : Prop :=
   0 ∈ chargeYukawaTop (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHu
 
@@ -260,15 +283,80 @@ instance : Decidable 𝓜.HasATopYukawa :=
 /-- The condition on the matter content for there to exist at least one copy of the coupling
 - `λᵇᵢⱼ 10ⁱ 5̄Mʲ 5̄Hd`
 -/
-def HasABottomYukawa (𝓜 : MatterContent I) : Prop := ∃ ti ∈ 𝓜.quantaTen,
-  ∃ fj ∈ 𝓜.quantaBarFiveMatter,
-  ti.q.1 + fj.q.1 + 𝓜.qHd.1 = 0
+def HasABottomYukawa (𝓜 : MatterContent I) : Prop :=
+  0 ∈ chargeYukawaBottom (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
+    (𝓜.quantaTen.map QuantaTen.q) 𝓜.qHu
 
 instance : Decidable 𝓜.HasABottomYukawa :=
-  haveI : DecidablePred fun (ti : QuantaTen I) =>
-      ∃ fj ∈ 𝓜.quantaBarFiveMatter, ti.q.1 + fj.q.1 + 𝓜.qHd.1 = 0 := fun _ =>
-        Multiset.decidableExistsMultiset
-  Multiset.decidableExistsMultiset
+  Multiset.decidableMem _ _
+
+/-!
+
+## More sophisticated checks
+-/
+
+lemma lambdaTerm_K1Term_W1Term_subset_check {I : CodimensionOneConfig} {n : ℕ} (𝓜 : MatterContent I)
+    (hcard : 𝓜.quantaBarFiveMatter.card = n) (h : 𝓜.ProtonDecayU1Constrained)
+    (S : Multiset (I.allowedTenCharges))
+    (hS : ∀ (F : Finset { x // x ∈ I.allowedBarFiveCharges }), F.card = n →
+      F ⊆ Finset.univ → F.card = n →
+      (0 ∈ chargeW1Term F.val S ∨ 0 ∈ chargeLambdaTerm F.val S) ∨ 0 ∈ chargeK1Term F.val S:= by
+      decide) :
+      ¬ S ⊆ 𝓜.quantaTen.map QuantaTen.q := by
+  intro hn
+  have hL1 := chargeLambdaTerm_subset_q10 (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
+    (𝓜.quantaTen.map QuantaTen.q) h.2.1 _ hn
+  have hW1 := chargeW1Term_subset_q10 (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
+    (𝓜.quantaTen.map QuantaTen.q) h.1 _ hn
+  have hK1 := chargeK1Term_subset_q10 (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
+    (𝓜.quantaTen.map QuantaTen.q) h.2.2.2 _ hn
+  apply not_or_intro (not_or_intro hW1 hL1) hK1
+  have h5 : ((𝓜.quantaBarFiveMatter).map QuantaBarFive.q).card = n := by
+    rw [Multiset.card_map]
+    exact hcard
+  rw [𝓜.quantaBarFiveMatter_map_q_eq_toFinset] at h5 ⊢
+  generalize (𝓜.quantaBarFiveMatter.map QuantaBarFive.q).toFinset = F at h5 ⊢
+  have hW1T : F ∈ (Finset.powerset (Finset.univ)).filter (fun x => x.card = n) := by
+    rw [Finset.mem_filter]
+    rw [Finset.mem_powerset]
+    simp_all only [Finset.card_val, and_true]
+    exact Finset.subset_univ F
+  revert F
+  simp only [Finset.card_val, Finset.univ_eq_attach, Finset.mem_filter, Finset.mem_powerset,
+    Int.reduceNeg, and_imp]
+  exact hS
+
+lemma lambdaTerm_K1Term_W1Term_singleton_check {I : CodimensionOneConfig} {n : ℕ}
+    (𝓜 : MatterContent I)
+    (hcard : 𝓜.quantaBarFiveMatter.card = n) (h : 𝓜.ProtonDecayU1Constrained)
+    (a : (I.allowedTenCharges))
+    (ha : ∀ (F : Finset { x // x ∈ I.allowedBarFiveCharges }), F.card = n →
+      F ⊆ Finset.univ → F.card = n →
+      (0 ∈ chargeW1Term F.val {a} ∨ 0 ∈ chargeLambdaTerm F.val {a}) ∨
+      0 ∈ chargeK1Term F.val {a} := by decide) :
+    a ∉ 𝓜.quantaTen.map QuantaTen.q := by
+  intro hn
+  have hL1 := chargeLambdaTerm_single_q10 (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
+    (𝓜.quantaTen.map QuantaTen.q) h.2.1 _ hn
+  have hW1 := chargeW1Term_single_q10 (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
+    (𝓜.quantaTen.map QuantaTen.q) h.1 _ hn
+  have hK1 := chargeK1Term_single_q10 (𝓜.quantaBarFiveMatter.map QuantaBarFive.q)
+    (𝓜.quantaTen.map QuantaTen.q) h.2.2.2 _ hn
+  apply not_or_intro (not_or_intro hW1 hL1) hK1
+  have h5 : ((𝓜.quantaBarFiveMatter).map QuantaBarFive.q).card = n := by
+    rw [Multiset.card_map]
+    exact hcard
+  rw [𝓜.quantaBarFiveMatter_map_q_eq_toFinset] at h5 ⊢
+  generalize (𝓜.quantaBarFiveMatter.map QuantaBarFive.q).toFinset = F at h5 ⊢
+  have hW1T : F ∈ (Finset.powerset (Finset.univ)).filter (fun x => x.card = n) := by
+    rw [Finset.mem_filter]
+    rw [Finset.mem_powerset]
+    simp_all only [Finset.card_val, and_true]
+    exact Finset.subset_univ F
+  revert F
+  simp only [Finset.card_val, Finset.univ_eq_attach, Finset.mem_filter, Finset.mem_powerset,
+    Int.reduceNeg, and_imp]
+  exact ha
 
 end MatterContent
 end SU5U1

From 1870115ea4bc235b013808bcf90dc056667a6942 Mon Sep 17 00:00:00 2001
From: jstoobysmith <72603918+jstoobysmith@users.noreply.github.com>
Date: Fri, 2 May 2025 07:22:42 +0000
Subject: [PATCH 21/32] chore: Fix file deletion

---
 .../TightBindingChain/Basic.lean              | 24 +++++++++++++++++++
 1 file changed, 24 insertions(+)
 create mode 100644 PhysLean/CondensedMatter/TightBindingChain/Basic.lean

diff --git a/PhysLean/CondensedMatter/TightBindingChain/Basic.lean b/PhysLean/CondensedMatter/TightBindingChain/Basic.lean
new file mode 100644
index 000000000..ee8670625
--- /dev/null
+++ b/PhysLean/CondensedMatter/TightBindingChain/Basic.lean
@@ -0,0 +1,24 @@
+/-
+Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import PhysLean.Meta.TODO.Basic
+/-!
+
+# The tight binding chain
+
+The tight binding chain corresponds to an electron in motion
+in a 1d solid with the assumption the electron can sit only on the atoms of the solid.
+
+The solid is assumed to consist of `N` sites with a seperation of `a` between them.
+
+## Refs.
+
+- https://www.damtp.cam.ac.uk/user/tong/aqm/aqmtwo.pdf
+
+-/
+
+TODO "BBZAB" "Prove results related to the one-dimensional tight binding chain.
+    This is related to the following issue/feature-request:
+    https://github.com/HEPLean/PhysLean/issues/523 "

From a67a6ab56d5eb4d5cfcb0c159b2e04f8424b0d67 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Sat, 3 May 2025 12:04:44 +0200
Subject: [PATCH 22/32] update Pseudo-Riemannian

fixed last review suggestions
---
 .../Metric/PseudoRiemannian/Defs.lean         | 141 +++++++-----------
 .../Geometry/Metric/Riemannian/Defs.lean      |  41 ++---
 2 files changed, 69 insertions(+), 113 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index a020f5289..d72f6cb6b 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -3,15 +3,12 @@ Copyright (c) 2025 Matteo Cipollina. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Matteo Cipollina
 -/
-
 import Mathlib.Analysis.InnerProductSpace.Basic
 import Mathlib.Analysis.RCLike.Lemmas
 import Mathlib.Geometry.Manifold.MFDeriv.Defs
-import Mathlib.Geometry.Manifold.VectorBundle.Basic
 import Mathlib.LinearAlgebra.BilinearForm.Properties
 import Mathlib.LinearAlgebra.QuadraticForm.Real
 import Mathlib.Topology.LocallyConstant.Basic
-
 /-!
 # Pseudo-Riemannian Metrics on Smooth Manifolds
 
@@ -52,7 +49,7 @@ open scoped Manifold Bundle LinearMap Dual
 
 namespace QuadraticForm
 
-variable {K : Type*} [Field K] [LinearOrder K]
+variable {K : Type*} [Field K]
 
 /-- The negative dimension (or index) of a quadratic form is the dimension
     of a maximal negative definite subspace. -/
@@ -153,7 +150,7 @@ theorem rankNeg_eq_zero {E : Type*} [AddCommGroup E]
 end QuadraticForm
 
 /-- Helper function to convert the metric tensor at `x` to a quadratic form. -/
-private def PseudoRiemannianMetricValToQuadraticForm
+private def pseudoRiemannianMetricValToQuadraticForm
     {E : Type v} [NormedAddCommGroup E] [NormedSpace ℝ E]
     {H : Type w} [TopologicalSpace H]
     {M : Type w} [TopologicalSpace M] [ChartedSpace H M]
@@ -185,17 +182,12 @@ structure PseudoRiemannianMetric
     (E : Type v) (H : Type w) (M : Type w) (n : WithTop ℕ∞)
     [inst_norm_grp_E : NormedAddCommGroup E]
     [inst_norm_sp_E : NormedSpace ℝ E]
-    [inst_findim_E : FiniteDimensional ℝ E]
     [inst_top_H : TopologicalSpace H]
     [inst_top_M : TopologicalSpace M]
     [inst_chart_M : ChartedSpace H M]
     [inst_chart_E : ChartedSpace H E]
     (I : ModelWithCorners ℝ E H)
     [inst_mani : IsManifold I (n + 1) M]
-    [inst_total_top : TopologicalSpace (TotalSpace E (TangentSpace I : M → Type _))]
-    [inst_fb : FiberBundle E (TangentSpace I : M → Type _)]
-    [inst_vb : VectorBundle ℝ E (TangentSpace I : M → Type _)]
-    [inst_cmvb : ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
     [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)] :
       Type (max u v w) where
   /-- The metric tensor at each point `x : M`, represented as a continuous linear map
@@ -219,27 +211,18 @@ structure PseudoRiemannianMetric
   protected negDim_isLocallyConstant :
     IsLocallyConstant (fun x : M =>
       have : FiniteDimensional ℝ (TangentSpace I x) := inferInstance
-      (PseudoRiemannianMetricValToQuadraticForm val symm x).negDim)
+      (pseudoRiemannianMetricValToQuadraticForm val symm x).negDim)
 
 namespace PseudoRiemannianMetric
 
 variable {E : Type v} {H : Type w} {M : Type w} {n : WithTop ℕ∞}
-variable [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+variable [NormedAddCommGroup E] [NormedSpace ℝ E]
 variable [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E]
 variable {I : ModelWithCorners ℝ E H}
 variable [IsManifold I (n + 1) M]
-variable [TopologicalSpace (TotalSpace E (TangentSpace I : M → Type _))]
-variable [FiberBundle E (TangentSpace I : M → Type _)]
-variable [VectorBundle ℝ E (TangentSpace I : M → Type _)]
-variable [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
 variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
 variable {g : PseudoRiemannianMetric E H M n I}
 
-instance TangentSpace.addCommGroup (x : M) : AddCommGroup (TangentSpace I x) := by infer_instance
-instance TangentSpace.module (x : M) : Module ℝ (TangentSpace I x) := by infer_instance
-instance TangentSpace.finiteDimensional (x : M) : FiniteDimensional ℝ (TangentSpace I x) := by
-  infer_instance
-
 /-- Convert the metric's continuous linear map representation `val x` to the algebraic
     `LinearMap.BilinForm`. -/
 def toBilinForm (g : PseudoRiemannianMetric E H M n I) (x : M) :
@@ -260,43 +243,33 @@ def toBilinForm (g : PseudoRiemannianMetric E H M n I) (x : M) :
 /-- Convert a pseudo-Riemannian metric at a point `x` to a quadratic form `v ↦ gₓ(v, v)`. -/
 def toQuadraticForm (g : PseudoRiemannianMetric E H M n I) (x : M) :
     QuadraticForm ℝ (TangentSpace I x) :=
-  PseudoRiemannianMetricValToQuadraticForm g.val g.symm x
+  pseudoRiemannianMetricValToQuadraticForm g.val g.symm x
 
 -- Coercion from PseudoRiemannianMetric to its function representation.
 instance coeFunInst : CoeFun (PseudoRiemannianMetric E H M n I)
         (fun _ => ∀ x : M, TangentSpace I x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ)) where
    coe g := g.val
 
-@[simp] lemma toBilinForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M)
+@[simp]
+lemma toBilinForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M)
     (v w : TangentSpace I x) :
   toBilinForm g x v w = g.val x v w := rfl
 
-@[simp] lemma toQuadraticForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M)
+@[simp]
+lemma toQuadraticForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M)
     (v : TangentSpace I x) :
   toQuadraticForm g x v = g.val x v v := rfl
 
-@[simp] lemma toBilinForm_isSymm (g : PseudoRiemannianMetric E H M n I) (x : M) :
+@[simp]
+lemma toBilinForm_isSymm (g : PseudoRiemannianMetric E H M n I) (x : M) :
     (toBilinForm g x).IsSymm := by
   intro v w; simp only [toBilinForm_apply]; exact g.symm x v w
 
-@[simp] lemma toBilinForm_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x : M) :
+@[simp]
+lemma toBilinForm_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x : M) :
     (toBilinForm g x).Nondegenerate := by
   intro v hv; simp_rw [toBilinForm_apply] at hv; exact g.nondegenerate x v hv
 
-lemma symm' (x : M) (v w : TangentSpace I x) : (g.val x v) w = (g.val x w) v :=
-  g.symm x v w
-
-lemma nondegenerate' (x : M) (v : TangentSpace I x)
-    (h : ∀ w : TangentSpace I x, (g.val x v) w = 0) : v = 0 := g.nondegenerate x v h
-
-lemma smooth' (x₀ : M) (v w : E) :
-    ContDiffWithinAt ℝ n (fun y => g.val ((extChartAt I x₀).symm y)
-    (mfderiv I I ((extChartAt I x₀).symm) y v) (mfderiv I I ((extChartAt I x₀).symm) y w))
-    ((extChartAt I x₀).target) ((extChartAt I x₀) x₀) := g.smooth_in_charts' x₀ v w
-
-lemma negDim_isLocallyConstant' : IsLocallyConstant (fun x => (toQuadraticForm g x).negDim) :=
-  g.negDim_isLocallyConstant
-
 /-- The "musical" isomorphism (index lowering) from the tangent space to its dual,
     induced by a pseudo-Riemannian metric. -/
 def flat (g : PseudoRiemannianMetric E H M n I) (x : M) :
@@ -305,26 +278,30 @@ def flat (g : PseudoRiemannianMetric E H M n I) (x : M) :
     map_add' := λ v w => by simp only [ContinuousLinearMap.map_add],
     map_smul' := λ a v => by simp only [ContinuousLinearMap.map_smul]; rfl }
 
-@[simp] lemma flat_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
+@[simp]
+lemma flat_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
   (flat g x v) w = g.val x v w := by rfl
 
 /-- The musical isomorphism as a continuous linear map. -/
 def flatL (g : PseudoRiemannianMetric E H M n I) (x : M) :
-    TangentSpace I x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ) :=
-  { toFun := λ v => g.val x v,
-    map_add' := λ v w => by simp only [ContinuousLinearMap.map_add],
-    map_smul' := λ a v => by simp only [ContinuousLinearMap.map_smul]; rfl,
-    cont := ContinuousLinearMap.continuous (g.val x) }
-
-@[simp] lemma flatL_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
+    TangentSpace I x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ) where
+  toFun := λ v => g.val x v
+  map_add' := λ v w => by simp only [ContinuousLinearMap.map_add]
+  map_smul' := λ a v => by simp only [ContinuousLinearMap.map_smul]; rfl
+  cont := ContinuousLinearMap.continuous (g.val x)
+
+@[simp]
+lemma flatL_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
   (flatL g x v) w = g.val x v w := rfl
 
-@[simp] lemma flat_inj (g : PseudoRiemannianMetric E H M n I) (x : M) :
+@[simp]
+lemma flat_inj (g : PseudoRiemannianMetric E H M n I) (x : M) :
     Function.Injective (flat g x) := by
   rw [← LinearMap.ker_eq_bot]; apply LinearMap.ker_eq_bot'.mpr
-  intro v hv; apply g.nondegenerate' x v; intro w; exact DFunLike.congr_fun hv w
+  intro v hv; apply g.nondegenerate x v; intro w; exact DFunLike.congr_fun hv w
 
-@[simp] lemma flatL_inj (g : PseudoRiemannianMetric E H M n I) (x : M) :
+@[simp]
+lemma flatL_inj (g : PseudoRiemannianMetric E H M n I) (x : M) :
     Function.Injective (flatL g x) :=
   flat_inj g x
 
@@ -344,12 +321,6 @@ def ContinuousLinearMap.equivModuleDual (𝕜 E : Type*) [NontriviallyNormedFiel
   left_inv f := by ext; rfl
   right_inv φ := rfl
 
-/-- Helper theorem: in finite dimensions, every linear map is continuous -/
-theorem continuous_linear_map_of_finite_dimension {𝕜 E : Type*} [NontriviallyNormedField 𝕜]
-    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] [CompleteSpace 𝕜]
-    (f : E →ₗ[𝕜] 𝕜) : Continuous f :=
-  by exact LinearMap.continuous_of_finiteDimensional f
-
 /-- For a finite-dimensional normed space, the dimension of the continuous dual
 equals the dimension of the original space. -/
 lemma finrank_continuousDual_eq_finrank {𝕜 E : Type*} [NontriviallyNormedField 𝕜]
@@ -361,10 +332,11 @@ lemma finrank_continuousDual_eq_finrank {𝕜 E : Type*} [NontriviallyNormedFiel
     exact finrank_linearMap_self 𝕜 𝕜 E
   rw [LinearEquiv.finrank_eq h1, h2]
 
-@[simp] lemma flatL_surj
+@[simp]
+lemma flatL_surj
     (g : PseudoRiemannianMetric E H M n I) (x : M) :
     Function.Surjective (g.flatL x) := by
-  haveI : FiniteDimensional ℝ (TangentSpace I x) := TangentSpace.finiteDimensional x
+  haveI : FiniteDimensional ℝ (TangentSpace I x) := inst_tangent_findim x
   have h_finrank_eq : finrank ℝ (TangentSpace I x) = finrank ℝ (TangentSpace I x →L[ℝ] ℝ) := by
     have h_dual_eq : finrank ℝ (TangentSpace I x →L[ℝ] ℝ) = finrank ℝ (Module.Dual ℝ
     (TangentSpace I x)) := by
@@ -390,7 +362,7 @@ lemma finrank_continuousDual_eq_finrank {𝕜 E : Type*} [NontriviallyNormedFiel
 
 /-- The "musical" isomorphism (index lowering) from the tangent space to its dual,
     as a continuous linear equivalence. -/
-@[simp] def flatEquiv
+def flatEquiv
     (g : PseudoRiemannianMetric E H M n I)
     (x : M) :
     TangentSpace I x ≃L[ℝ] (TangentSpace I x →L[ℝ] ℝ) :=
@@ -403,7 +375,8 @@ lemma coe_flatEquiv
     (g : PseudoRiemannianMetric E H M n I) (x : M) :
     (g.flatEquiv x : TangentSpace I x →ₗ[ℝ] (TangentSpace I x →L[ℝ] ℝ)) = g.flatL x := rfl
 
-@[simp] lemma flatEquiv_apply
+@[simp]
+lemma flatEquiv_apply
     (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
     (g.flatEquiv x v) w = g.val x v w := rfl
 
@@ -432,17 +405,20 @@ noncomputable def sharp
     (g : PseudoRiemannianMetric E H M n I) (x : M) :
     (TangentSpace I x →L[ℝ] ℝ) →ₗ[ℝ] TangentSpace I x := (g.sharpEquiv x).toLinearEquiv.toLinearMap
 
-@[simp] lemma sharpL_apply_flatL
+@[simp]
+lemma sharpL_apply_flatL
      (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) :
     g.sharpL x (g.flatL x v) = v :=
   (g.flatEquiv x).left_inv v
 
-@[simp] lemma flatL_apply_sharpL
+@[simp]
+lemma flatL_apply_sharpL
     (g : PseudoRiemannianMetric E H M n I) (x : M) (ω : TangentSpace I x →L[ℝ] ℝ) :
     g.flatL x (g.sharpL x ω) = ω := (g.flatEquiv x).right_inv ω
 
 /-- Applying `sharp` then `flat` recovers the original covector. -/
-@[simp] lemma flat_sharp_apply
+@[simp]
+lemma flat_sharp_apply
     (g : PseudoRiemannianMetric E H M n I) (x : M) (ω : TangentSpace I x →L[ℝ] ℝ) :
     g.flat x (g.sharp x ω) = ω := by
   have := flatL_apply_sharpL g x ω
@@ -450,7 +426,8 @@ noncomputable def sharp
              ContinuousLinearEquiv.coe_coe, LinearEquiv.coe_coe] at this ⊢
   exact this
 
-@[simp] lemma sharp_flat_apply
+@[simp]
+lemma sharp_flat_apply
     (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) :
     g.sharp x (g.flat x v) = v := by
   have := sharpL_apply_flatL g x v
@@ -459,7 +436,8 @@ noncomputable def sharp
   exact this
 
 /-- The metric evaluated at `sharp ω₁` and `sharp ω₂`. -/
-@[simp] lemma apply_sharp_sharp
+@[simp]
+lemma apply_sharp_sharp
     (g : PseudoRiemannianMetric E H M n I) (x : M) (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
     g.val x (g.sharpL x ω₁) (g.sharpL x ω₂) = ω₁ (g.sharpL x ω₂) := by
   rw [← flatL_apply g x (g.sharpL x ω₁)]
@@ -470,33 +448,26 @@ lemma apply_vec_sharp
     (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x)
     (ω : TangentSpace I x →L[ℝ] ℝ) :
     g.val x v (g.sharpL x ω) = ω v := by
-  rw [g.symm' x v (g.sharpL x ω)]
+  rw [g.symm x v (g.sharpL x ω)]
   rw [← flatL_apply g x (g.sharpL x ω)]
   rw [flatL_apply_sharpL g x ω]
 
-end PseudoRiemannianMetric
-
-namespace PseudoRiemannianMetric
+section Cotangent
 
 variable {E : Type v} {H : Type w} {M : Type w} {n : WithTop ℕ∞}
-variable [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+variable [NormedAddCommGroup E] [NormedSpace ℝ E]
 variable [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E]
 variable {I : ModelWithCorners ℝ E H}
 variable [IsManifold I (n + 1) M]
-variable [TopologicalSpace (TotalSpace E (TangentSpace I : M → Type _))]
-variable [FiberBundle E (TangentSpace I : M → Type _)]
-variable [VectorBundle ℝ E (TangentSpace I : M → Type _)]
-variable [ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
 variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
 
-section Cotangent
-
 /-- The value of the induced metric on the cotangent space at point `x`. -/
 noncomputable def cotangentMetricVal (g : PseudoRiemannianMetric E H M n I) (x : M)
     (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) : ℝ :=
   g.val x (g.sharpL x ω₁) (g.sharpL x ω₂)
 
-@[simp] lemma cotangentMetricVal_eq_apply_sharp (g : PseudoRiemannianMetric E H M n I) (x : M)
+@[simp]
+lemma cotangentMetricVal_eq_apply_sharp (g : PseudoRiemannianMetric E H M n I) (x : M)
     (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
   cotangentMetricVal g x ω₁ ω₂ = ω₁ (g.sharpL x ω₂) := by
   rw [cotangentMetricVal, apply_sharp_sharp]
@@ -505,7 +476,7 @@ lemma cotangentMetricVal_symm (g : PseudoRiemannianMetric E H M n I) (x : M)
     (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
   cotangentMetricVal g x ω₁ ω₂ = cotangentMetricVal g x ω₂ ω₁ := by
   unfold cotangentMetricVal
-  rw [g.symm' x (g.sharpL x ω₁) (g.sharpL x ω₂)]
+  rw [g.symm x (g.sharpL x ω₁) (g.sharpL x ω₂)]
 
 /-- The induced metric on the cotangent space at point `x` as a bilinear form. -/
 noncomputable def cotangentToBilinForm (g : PseudoRiemannianMetric E H M n I) (x : M) :
@@ -559,15 +530,18 @@ noncomputable def cotangentToQuadraticForm (g : PseudoRiemannianMetric E H M n I
             simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.add_apply]
             ring⟩
 
-@[simp] lemma cotangentToBilinForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M)
+@[simp]
+lemma cotangentToBilinForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M)
     (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
   cotangentToBilinForm g x ω₁ ω₂ = cotangentMetricVal g x ω₁ ω₂ := rfl
 
-@[simp] lemma cotangentToQuadraticForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M)
+@[simp]
+lemma cotangentToQuadraticForm_apply (g : PseudoRiemannianMetric E H M n I) (x : M)
     (ω : TangentSpace I x →L[ℝ] ℝ) :
   cotangentToQuadraticForm g x ω = cotangentMetricVal g x ω ω := rfl
 
-@[simp] lemma cotangentToBilinForm_isSymm (g : PseudoRiemannianMetric E H M n I) (x : M) :
+@[simp]
+lemma cotangentToBilinForm_isSymm (g : PseudoRiemannianMetric E H M n I) (x : M) :
     (cotangentToBilinForm g x).IsSymm := by
   intro ω₁ ω₂; simp only [cotangentToBilinForm_apply]; exact cotangentMetricVal_symm g x ω₁ ω₂
 
@@ -590,7 +564,8 @@ lemma cotangentMetricVal_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x
     exact h_apply
   exact h_forall v
 
-@[simp] lemma cotangentToBilinForm_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x : M) :
+@[simp]
+lemma cotangentToBilinForm_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x : M) :
     (cotangentToBilinForm g x).Nondegenerate := by
   intro ω hω
   apply cotangentMetricVal_nondegenerate g x ω
diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index d3179c311..9c25ed298 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -3,24 +3,9 @@ Copyright (c) 2025 Matteo Cipollina. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Matteo Cipollina
 -/
-
-import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
-import Mathlib.LinearAlgebra.QuadraticForm.Dual
-
 import Mathlib.Algebra.Lie.OfAssociative
-import Mathlib.Analysis.InnerProductSpace.Basic
-import Mathlib.Analysis.Normed.Field.Instances
-import Mathlib.Analysis.RCLike.Lemmas
-import Mathlib.Data.Real.StarOrdered
-import Mathlib.Geometry.Manifold.MFDeriv.Defs
-import Mathlib.Geometry.Manifold.VectorBundle.Basic
-import Mathlib.LinearAlgebra.BilinearForm.Properties
-import Mathlib.LinearAlgebra.FreeModule.PID
-import Mathlib.LinearAlgebra.QuadraticForm.Real
-import Mathlib.RingTheory.Henselian
-import Mathlib.Topology.Algebra.Module.ModuleTopology
-import Mathlib.Topology.LocallyConstant.Basic
 import Mathlib.MeasureTheory.Integral.IntervalIntegral
+import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
 /-!
 # Riemannian Metric Definitions
 
@@ -35,7 +20,7 @@ open PseudoRiemannianMetric InnerProductSpace
 
 noncomputable section
 
-variable {E : Type v} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+variable {E : Type v} [NormedAddCommGroup E] [NormedSpace ℝ E]
 variable {H : Type w} [TopologicalSpace H]
 variable {M : Type w} [TopologicalSpace M] [ChartedSpace H M] [ChartedSpace H E]
 variable {I : ModelWithCorners ℝ E H} {n : ℕ∞}
@@ -46,10 +31,6 @@ variable {I : ModelWithCorners ℝ E H} {n : ℕ∞}
 structure RiemannianMetric
   (I : ModelWithCorners ℝ E H) (n : ℕ∞) (M : Type w)
   [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (n + 1) M]
-  [inst_top : TopologicalSpace (TangentBundle I M)]
-  [inst_fib : FiberBundle E (TangentSpace I : M → Type _)]
-  [inst_vec : VectorBundle ℝ E (TangentSpace I : M → Type _)]
-  [inst_cmvb : ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
   [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)] : Type _ where
   /-- The underlying pseudo-Riemannian metric. -/
   toPseudoRiemannianMetric : PseudoRiemannianMetric E H M (n) I
@@ -60,17 +41,14 @@ namespace RiemannianMetric
 
 variable {I : ModelWithCorners ℝ E H} {n : ℕ∞} {M : Type w}
 variable [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (n + 1) M]
-variable [inst_top : TopologicalSpace (TangentBundle I M)]
-variable [inst_fib : FiberBundle E (TangentSpace I : M → Type _)]
-variable [inst_vec : VectorBundle ℝ E (TangentSpace I : M → Type _)]
-variable [inst_cmvb : ContMDiffVectorBundle n E (TangentSpace I : M → Type _) I]
 variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
 
 /-- Coercion from RiemannianMetric to its underlying PseudoRiemannianMetric. -/
 instance : Coe (RiemannianMetric I n M) (PseudoRiemannianMetric E H M (n) I) where
   coe g := g.toPseudoRiemannianMetric
 
-@[simp] lemma pos_def (g : RiemannianMetric I n M) (x : M) (v : TangentSpace I x)
+@[simp]
+lemma pos_def (g : RiemannianMetric I n M) (x : M) (v : TangentSpace I x)
     (hv : v ≠ 0) :
   (g.toPseudoRiemannianMetric.val x v) v > 0 := g.pos_def' x v hv
 
@@ -87,17 +65,19 @@ abbrev toQuadraticForm (g : RiemannianMetric I n M) (x : M) :
   g.toPseudoRiemannianMetric.toQuadraticForm x
 
 /-- The quadratic form associated with a Riemannian metric is positive definite. -/
-@[simp] lemma toQuadraticForm_posDef (g : RiemannianMetric I n M) (x : M) :
+@[simp]
+lemma toQuadraticForm_posDef (g : RiemannianMetric I n M) (x : M) :
     (g.toQuadraticForm x).PosDef :=
   λ v hv => g.pos_def x v hv
 
 /-- The application of a Riemannian metric's quadratic form to a vector. -/
-@[simp] lemma toQuadraticForm_apply (g : RiemannianMetric I n M) (x : M)
+@[simp]
+lemma toQuadraticForm_apply (g : RiemannianMetric I n M) (x : M)
     (v : TangentSpace I x) :
     g.toQuadraticForm x v = g.toPseudoRiemannianMetric.val x v v := by
   simp only [toQuadraticForm, PseudoRiemannianMetric.toQuadraticForm_apply]
 
-theorem riemannian_metric_negDim_zero (g : RiemannianMetric I n M) (x : M) :
+lemma riemannian_metric_negDim_zero (g : RiemannianMetric I n M) (x : M) :
     (g.toQuadraticForm x).negDim = 0 := by
     apply QuadraticForm.rankNeg_eq_zero
     exact g.toQuadraticForm_posDef x
@@ -106,7 +86,8 @@ theorem riemannian_metric_negDim_zero (g : RiemannianMetric I n M) (x : M) :
 def inner (g : RiemannianMetric I n M) (x : M) (v w : TangentSpace I x) : ℝ :=
   g.toPseudoRiemannianMetric.val x v w
 
-@[simp] lemma inner_apply (g : RiemannianMetric I n M) (x : M) (v w : TangentSpace I x) :
+@[simp]
+lemma inner_apply (g : RiemannianMetric I n M) (x : M) (v w : TangentSpace I x) :
   inner g x v w = g.toPseudoRiemannianMetric.val x v w := rfl
 
 variable (g : RiemannianMetric I n M) (x : M)

From e10d58469bff4058507df2edd38055d8404bd6ae Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Sat, 3 May 2025 12:05:22 +0200
Subject: [PATCH 23/32] Update Defs.lean

---
 PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index 9c25ed298..4f0c2266e 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -96,7 +96,7 @@ variable (g : RiemannianMetric I n M) (x : M)
 lemma inner_symm (v w : TangentSpace I x) :
     g.inner x v w = g.inner x w v := by
   simp only [inner_apply]
-  exact g.toPseudoRiemannianMetric.symm' x v w
+  exact g.toPseudoRiemannianMetric.symm x v w
 
 /-- The inner product space core for the tangent space at a point, derived from the
 Riemannian metric. -/

From a86be209efc7a2f93f8f01b44f9a5bbaadf19cfd Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Sat, 3 May 2025 14:11:18 +0200
Subject: [PATCH 24/32] Update Defs.lean

---
 .../Metric/PseudoRiemannian/Defs.lean         | 20 +++++--------------
 1 file changed, 5 insertions(+), 15 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index d72f6cb6b..f2aa56d16 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -62,16 +62,6 @@ noncomputable def negDim {E : Type*} [AddCommGroup E]
   let w := Classical.choose h_exists
   exact Finset.card (Finset.filter (fun i => w i = SignType.neg) Finset.univ)
 
-/-- The i-th standard basis vector has a 1 in the i-th position. -/
-lemma Pi.basisFun_apply_same {𝕜 ι : Type*} [Field 𝕜] [Fintype ι] [DecidableEq ι] (i : ι) :
-    (Pi.basisFun 𝕜 ι) i i = 1 := by
-  simp only [Pi.basisFun_apply, Pi.single_eq_same]
-
-/-- The i-th standard basis vector has 0 in all positions j ≠ i. -/
-lemma Pi.basisFun_apply_ne {𝕜 ι : Type*} [Field 𝕜] [Fintype ι] [DecidableEq ι] (i j : ι)
-    (h : j ≠ i) : (Pi.basisFun 𝕜 ι) i j = 0 := by
-  simp only [Pi.basisFun_apply, Pi.single_eq_of_ne h]
-
 /-- For a standard basis vector in a weighted sum of squares, only one term in the sum
     is nonzero. -/
 lemma QuadraticMap.weightedSumSquares_basis_vector {E : Type*} [AddCommGroup E]
@@ -192,23 +182,23 @@ structure PseudoRiemannianMetric
       Type (max u v w) where
   /-- The metric tensor at each point `x : M`, represented as a continuous linear map
       `TₓM →L[ℝ] (TₓM →L[ℝ] ℝ)`. Applying it twice, `(val x v) w`, yields `gₓ(v, w)`. -/
-  protected val : ∀ (x : M), TangentSpace I x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ)
+  val : ∀ (x : M), TangentSpace I x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ)
   /-- The metric is symmetric: `gₓ(v, w) = gₓ(w, v)`. -/
-  protected symm : ∀ (x : M) (v w : TangentSpace I x), (val x v) w = (val x w) v
+  symm : ∀ (x : M) (v w : TangentSpace I x), (val x v) w = (val x w) v
   /-- The metric is non-degenerate: if `gₓ(v, w) = 0` for all `w`, then `v = 0`. -/
-  protected nondegenerate : ∀ (x : M) (v : TangentSpace I x), (∀ w : TangentSpace I x,
+  nondegenerate : ∀ (x : M) (v : TangentSpace I x), (∀ w : TangentSpace I x,
     (val x v) w = 0) → v = 0
   /-- The metric varies smoothly: Expressed in local coordinates via any chart `e`, the function
       `y ↦ g_{e.symm y}(mfderiv I I e.symm y v, mfderiv I I e.symm y w)` is `C^n` smooth on the
       chart's target `e.target` for any constant vectors `v, w` in the model space `E`. -/
-  protected smooth_in_charts' : ∀ (x₀ : M) (v w : E),
+  smooth_in_charts' : ∀ (x₀ : M) (v w : E),
     let e := extChartAt I x₀
     ContDiffWithinAt ℝ n
       (fun y => val (e.symm y) (mfderiv I I e.symm y v) (mfderiv I I e.symm y w))
       (e.target) (e x₀)
   /-- The negative dimension (`QuadraticForm.negDim`) of the metric's quadratic form is
       locally constant. On a connected manifold, this implies it is constant globally. -/
-  protected negDim_isLocallyConstant :
+  negDim_isLocallyConstant :
     IsLocallyConstant (fun x : M =>
       have : FiniteDimensional ℝ (TangentSpace I x) := inferInstance
       (pseudoRiemannianMetricValToQuadraticForm val symm x).negDim)

From 3e03d643851f7189b428b9cab46f9fc402318c54 Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Sat, 3 May 2025 15:33:10 +0200
Subject: [PATCH 25/32] fix lints

---
 PhysLean.lean                                             | 1 +
 .../Geometry/Metric/PseudoRiemannian/Defs.lean            | 6 +++---
 PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean | 8 +++++---
 3 files changed, 9 insertions(+), 6 deletions(-)

diff --git a/PhysLean.lean b/PhysLean.lean
index f362cfbf6..985001cdf 100644
--- a/PhysLean.lean
+++ b/PhysLean.lean
@@ -7,6 +7,7 @@ import PhysLean.ClassicalMechanics.Space.VectorIdentities
 import PhysLean.ClassicalMechanics.Time.Basic
 import PhysLean.ClassicalMechanics.VectorFields
 import PhysLean.CondensedMatter.Basic
+import PhysLean.CondensedMatter.TightBindingChain.Basic
 import PhysLean.Cosmology.Basic
 import PhysLean.Cosmology.FLRW.Basic
 import PhysLean.Electromagnetism.Basic
diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index f2aa56d16..ddf5ec1cb 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -270,7 +270,7 @@ def flat (g : PseudoRiemannianMetric E H M n I) (x : M) :
 
 @[simp]
 lemma flat_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
-  (flat g x v) w = g.val x v w := by rfl
+    (flat g x v) w = g.val x v w := by rfl
 
 /-- The musical isomorphism as a continuous linear map. -/
 def flatL (g : PseudoRiemannianMetric E H M n I) (x : M) :
@@ -282,7 +282,7 @@ def flatL (g : PseudoRiemannianMetric E H M n I) (x : M) :
 
 @[simp]
 lemma flatL_apply (g : PseudoRiemannianMetric E H M n I) (x : M) (v w : TangentSpace I x) :
-  (flatL g x v) w = g.val x v w := rfl
+    (flatL g x v) w = g.val x v w := rfl
 
 @[simp]
 lemma flat_inj (g : PseudoRiemannianMetric E H M n I) (x : M) :
@@ -397,7 +397,7 @@ noncomputable def sharp
 
 @[simp]
 lemma sharpL_apply_flatL
-     (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) :
+    (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x) :
     g.sharpL x (g.flatL x v) = v :=
   (g.flatEquiv x).left_inv v
 
diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index 4f0c2266e..18e0abeec 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -3,9 +3,11 @@ Copyright (c) 2025 Matteo Cipollina. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Matteo Cipollina
 -/
+
 import Mathlib.Algebra.Lie.OfAssociative
 import Mathlib.MeasureTheory.Integral.IntervalIntegral
 import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
+
 /-!
 # Riemannian Metric Definitions
 
@@ -79,8 +81,8 @@ lemma toQuadraticForm_apply (g : RiemannianMetric I n M) (x : M)
 
 lemma riemannian_metric_negDim_zero (g : RiemannianMetric I n M) (x : M) :
     (g.toQuadraticForm x).negDim = 0 := by
-    apply QuadraticForm.rankNeg_eq_zero
-    exact g.toQuadraticForm_posDef x
+  apply QuadraticForm.rankNeg_eq_zero
+  exact g.toQuadraticForm_posDef x
 
 /-- The inner product on the tangent space at point `x` induced by the Riemannian metric `g`. -/
 def inner (g : RiemannianMetric I n M) (x : M) (v w : TangentSpace I x) : ℝ :=
@@ -88,7 +90,7 @@ def inner (g : RiemannianMetric I n M) (x : M) (v w : TangentSpace I x) : ℝ :=
 
 @[simp]
 lemma inner_apply (g : RiemannianMetric I n M) (x : M) (v w : TangentSpace I x) :
-  inner g x v w = g.toPseudoRiemannianMetric.val x v w := rfl
+    inner g x v w = g.toPseudoRiemannianMetric.val x v w := rfl
 
 variable (g : RiemannianMetric I n M) (x : M)
 

From 21f522bea68644a7c500ffb3aeb9104792d9de7d Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Fri, 9 May 2025 19:22:31 +0200
Subject: [PATCH 26/32] Added Chart + modules

---
 PhysLean.lean                                 |   8 +
 PhysLean/Mathematics/Analysis/ContDiff.lean   | 230 ++++++++++++++++++
 .../Manifold/Chart/BilinearSmoothness.lean    | 135 ++++++++++
 .../Chart/CoordinateTransformations.lean      | 160 ++++++++++++
 .../Geometry/Manifold/Chart/Smoothness.lean   | 135 ++++++++++
 .../Geometry/Manifold/Chart/Utilities.lean    |  59 +++++
 .../Geometry/Manifold/PartialHomeomorph.lean  | 209 ++++++++++++++++
 .../Metric/PseudoRiemannian/Chart.lean        | 142 +++++++++++
 .../Metric/PseudoRiemannian/Defs.lean         |   8 +-
 .../LinearAlgebra/BilinearForm.lean           | 185 ++++++++++++++
 10 files changed, 1268 insertions(+), 3 deletions(-)
 create mode 100644 PhysLean/Mathematics/Analysis/ContDiff.lean
 create mode 100644 PhysLean/Mathematics/Geometry/Manifold/Chart/BilinearSmoothness.lean
 create mode 100644 PhysLean/Mathematics/Geometry/Manifold/Chart/CoordinateTransformations.lean
 create mode 100644 PhysLean/Mathematics/Geometry/Manifold/Chart/Smoothness.lean
 create mode 100644 PhysLean/Mathematics/Geometry/Manifold/Chart/Utilities.lean
 create mode 100644 PhysLean/Mathematics/Geometry/Manifold/PartialHomeomorph.lean
 create mode 100644 PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean
 create mode 100644 PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean

diff --git a/PhysLean.lean b/PhysLean.lean
index 985001cdf..1c4010a65 100644
--- a/PhysLean.lean
+++ b/PhysLean.lean
@@ -16,10 +16,18 @@ import PhysLean.Electromagnetism.FieldStrength.Derivative
 import PhysLean.Electromagnetism.Homogeneous
 import PhysLean.Electromagnetism.LorentzAction
 import PhysLean.Electromagnetism.MaxwellEquations
+import PhysLean.Mathematics.Analysis.ContDiff
 import PhysLean.Mathematics.FDerivCurry
 import PhysLean.Mathematics.Fin
 import PhysLean.Mathematics.Fin.Involutions
+import PhysLean.Mathematics.LinearAlgebra.BilinearForm
+import PhysLean.Mathematics.Geometry.Manifold.PartialHomeomorph
+import PhysLean.Mathematics.Geometry.Manifold.Chart.Utilities
+import PhysLean.Mathematics.Geometry.Manifold.Chart.BilinearSmoothness
+import PhysLean.Mathematics.Geometry.Manifold.Chart.Smoothness
+import PhysLean.Mathematics.Geometry.Manifold.Chart.CoordinateTransformations
 import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
+import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Chart
 import PhysLean.Mathematics.Geometry.Metric.Riemannian.Defs
 import PhysLean.Mathematics.LinearMaps
 import PhysLean.Mathematics.List
diff --git a/PhysLean/Mathematics/Analysis/ContDiff.lean b/PhysLean/Mathematics/Analysis/ContDiff.lean
new file mode 100644
index 000000000..f4438c13f
--- /dev/null
+++ b/PhysLean/Mathematics/Analysis/ContDiff.lean
@@ -0,0 +1,230 @@
+/-
+Copyright (c) 2025 Matteo Cipollina. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Matteo Cipollina
+-/
+
+import Mathlib.Analysis.Calculus.ContDiff.Operations
+import Mathlib.LinearAlgebra.Dual.Lemmas
+import Mathlib.LinearAlgebra.FreeModule.PID
+import Mathlib.RingTheory.Henselian
+import PhysLean.Mathematics.LinearAlgebra.BilinearForm
+
+/-!
+# Smoothness (ContDiff) Utilities
+
+This file provides utility lemmas and constructions for working with smooth
+functions (`ContDiff`) and continuity in the context of normed and finite-dimensional
+vector spaces over a nontrivially normed field.
+
+-/
+
+namespace ContDiff
+
+variable {𝕜 X V : Type*} [NontriviallyNormedField 𝕜]
+variable [NormedAddCommGroup X] [NormedSpace 𝕜 X]
+variable [NormedAddCommGroup V] [NormedSpace 𝕜 V]
+variable {f : X → V} {s : Set X} {x₀ : X} {n : WithTop ℕ∞}
+
+-- First direction: if f is C^n, then φ ∘ f is C^n for any continuous linear functional φ
+lemma comp_continuous_linear_apply_right
+    (hf : ContDiffWithinAt 𝕜 n f s x₀) (φ : V →L[𝕜] 𝕜) :
+    ContDiffWithinAt 𝕜 n (φ ∘ f) s x₀ :=
+  ContDiffWithinAt.comp x₀ φ.contDiff.contDiffWithinAt hf (Set.mapsTo_univ _ _)
+
+-- Second direction: in finite dimensions, if all projections are C^n, then f is C^n
+lemma of_forall_coord [FiniteDimensional 𝕜 V] [CompleteSpace 𝕜]
+    (h : ∀ φ : V →L[𝕜] 𝕜, ContDiffWithinAt 𝕜 n (φ ∘ f) s x₀) :
+    ContDiffWithinAt 𝕜 n f s x₀ := by
+  let b := Module.finBasis 𝕜 V
+  let equiv := b.equivFunL
+  suffices ContDiffWithinAt 𝕜 n (equiv ∘ f) s x₀ by
+    have hequiv_symm_smooth : ContDiff 𝕜 ⊤ equiv.symm := ContinuousLinearEquiv.contDiff equiv.symm
+    have hequiv_symm_smooth_n : ContDiff 𝕜 n equiv.symm :=
+      ContDiff.of_le hequiv_symm_smooth (le_top : n ≤ ⊤)
+    have h_eq : f = equiv.symm ∘ (equiv ∘ f) := by
+      ext x; simp only [Function.comp_apply, ContinuousLinearEquiv.symm_apply_apply];
+    rw [h_eq]
+    apply ContDiffWithinAt.comp x₀ hequiv_symm_smooth_n.contDiffWithinAt this (Set.mapsTo_univ _ _)
+  apply contDiffWithinAt_pi.mpr
+  intro i
+  let coord_i : V →L[𝕜] 𝕜 := LinearMap.toContinuousLinearMap (b.coord i)
+  exact h coord_i
+
+-- Full bidirectional lemma
+lemma iff_forall_coord [FiniteDimensional 𝕜 V] [CompleteSpace 𝕜] :
+    ContDiffWithinAt 𝕜 n f s x₀ ↔
+    ∀ φ : V →L[𝕜] 𝕜, ContDiffWithinAt 𝕜 n (φ ∘ f) s x₀ := by
+  constructor
+  · exact comp_continuous_linear_apply_right
+  · exact of_forall_coord
+
+end ContDiff
+
+section ContinuityBounds
+
+variable {𝕜 E F FHom : Type*} [NormedField 𝕜]
+
+@[simp]
+lemma AddMonoidHomClass.coe_fn_to_addMonoidHom
+    [FunLike FHom E F] [AddZeroClass E] [AddZeroClass F]
+    [AddMonoidHomClass FHom E F] (φ : FHom) :
+    ⇑(AddMonoidHomClass.toAddMonoidHom φ) = ⇑φ := by
+  rfl
+
+variable [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+
+/-- A bounded additive map is continuous at zero. -/
+lemma AddMonoidHom.continuousAt_zero_of_bound
+    (φ : AddMonoidHom E F) {C : ℝ} (h : ∀ x, ‖φ x‖ ≤ C * ‖x‖) :
+    ContinuousAt φ 0 := by
+  rw [Metric.continuousAt_iff]
+  intro ε εpos
+  simp only [map_zero φ, dist_zero_right]
+  by_cases hE : Subsingleton E
+  · use 1
+    refine ⟨zero_lt_one, fun y _hy_norm_lt_one => ?_⟩
+    rw [@Subsingleton.elim E hE y 0, map_zero φ, norm_zero]
+    exact εpos
+  · have C_nonneg : 0 ≤ C := by
+      obtain ⟨x_ne, y_ne, h_x_ne_y⟩ : ∃ x y : E, x ≠ y := by
+        contrapose! hE; exact { allEq := hE }
+      let z := x_ne - y_ne
+      have hz_ne_zero : z ≠ 0 := sub_ne_zero_of_ne h_x_ne_y
+      have hz_norm_pos : 0 < ‖z‖ := norm_pos_iff.mpr hz_ne_zero
+      by_contra hC_is_neg
+      push_neg at hC_is_neg
+      have h_phi_z_bound := h z
+      have H1 : 0 ≤ C * ‖z‖ := le_trans (norm_nonneg (φ z)) h_phi_z_bound
+      have H2 : C * ‖z‖ < 0 := mul_neg_of_neg_of_pos hC_is_neg hz_norm_pos
+      linarith [H1, H2]
+    by_cases hC_eq_zero : C = 0
+    · have phi_is_zero : φ = 0 := by
+        ext x_val
+        have h_phi_x_val_bound := h x_val
+        rw [hC_eq_zero, zero_mul] at h_phi_x_val_bound
+        exact norm_le_zero_iff.mp h_phi_x_val_bound
+      use 1
+      refine ⟨zero_lt_one, fun y _hy_norm_lt_one => ?_⟩
+      rw [phi_is_zero, AddMonoidHom.zero_apply, norm_zero]
+      exact εpos
+    · have C_pos : 0 < C := lt_of_le_of_ne C_nonneg fun a => hC_eq_zero (_root_.id (Eq.symm a))
+      use ε / C
+      refine ⟨div_pos εpos C_pos, fun y hy_norm_lt_delta => ?_⟩
+      calc
+        ‖φ y‖ ≤ C * ‖y‖ := h y
+        _ < C * (ε / C) := mul_lt_mul_of_pos_left hy_norm_lt_delta C_pos
+        _ = ε := by rw [mul_div_cancel₀ ε hC_eq_zero]
+
+omit [NormedSpace 𝕜 F] in
+/-- A semi-linear map that is linearly bounded by the norm of its input is continuous. -/
+lemma SemilinearMapClass.continuous_of_bound {𝕜₂ : Type*} [NormedField 𝕜₂] [NormedSpace 𝕜₂ F]
+    [FunLike FHom E F] {σ : 𝕜 →+* 𝕜₂} [SemilinearMapClass FHom σ E F]
+    {φ : FHom} {C : ℝ} (h : ∀ x, ‖φ x‖ ≤ C * ‖x‖) : Continuous φ := by
+  haveI : AddMonoidHomClass FHom E F := inferInstance
+  let φ_add_hom : AddMonoidHom E F := AddMonoidHomClass.toAddMonoidHom φ
+  exact continuous_of_continuousAt_zero φ_add_hom
+    (AddMonoidHom.continuousAt_zero_of_bound φ_add_hom h)
+
+/-- A function that is linearly bounded by the norm of its input is continuous. -/
+lemma AddMonoidHomClass.continuous_of_bound' [FunLike FHom E F] [AddMonoidHomClass FHom E F]
+    {φ : FHom} {C : ℝ} (h : ∀ x, ‖φ x‖ ≤ C * ‖x‖) : Continuous φ := by
+  let φ_add_hom : AddMonoidHom E F := AddMonoidHomClass.toAddMonoidHom φ
+  exact continuous_of_continuousAt_zero φ_add_hom
+    (AddMonoidHom.continuousAt_zero_of_bound φ_add_hom h)
+
+end ContinuityBounds
+
+namespace ContinuousLinearMap
+
+variable {X₁ E₁ F₁ G₁ E₁' F₁' : Type*} [NontriviallyNormedField 𝕜₁]
+  [NormedAddCommGroup X₁] [NormedSpace 𝕜₁ X₁]
+  [NormedAddCommGroup E₁] [NormedSpace 𝕜₁ E₁]
+  [NormedAddCommGroup F₁] [NormedSpace 𝕜₁ F₁]
+  [NormedAddCommGroup G₁] [NormedSpace 𝕜₁ G₁]
+  [NormedAddCommGroup E₁'] [NormedSpace 𝕜₁ E₁']
+  [NormedAddCommGroup F₁'] [NormedSpace 𝕜₁ F₁']
+  {n₁ : WithTop ℕ∞}
+
+/-- The `ContinuousLinearMap.bilinearComp` operation is smooth.
+    Given smooth functions `f : X₁ → (E₁ →L[𝕜₁] F₁ →L[𝕜₁] G₁)`, `g : X₁ → (E₁' →L[𝕜₁] E₁)`,
+    and `h : X₁ → (F₁' →L[𝕜₁] F₁)`, the composition `x ↦ (f x).bilinearComp (g x) (h x)`
+    is smooth. -/
+lemma contDiff_bilinearComp
+    {f : X₁ → E₁ →L[𝕜₁] F₁ →L[𝕜₁] G₁} {g : X₁ → E₁' →L[𝕜₁] E₁} {h : X₁ → F₁' →L[𝕜₁] F₁}
+    (hf : ContDiff 𝕜₁ n₁ f) (hg : ContDiff 𝕜₁ n₁ g) (hh : ContDiff 𝕜₁ n₁ h) :
+    ContDiff 𝕜₁ n₁ fun x => (f x).bilinearComp (g x) (h x) := by
+  have h1 : ContDiff 𝕜₁ n₁ (fun x ↦ (f x).comp (g x)) := ContDiff.clm_comp hf hg
+  let L_flip1 := ContinuousLinearMap.flipₗᵢ 𝕜₁ E₁' F₁ G₁
+  have eq_flip : ∀ x, L_flip1 ((f x).comp (g x)) = ((f x).comp (g x)).flip := by
+    intro x
+    rfl
+  have h2 : ContDiff 𝕜₁ n₁ (fun x => ((f x).comp (g x)).flip) := by
+    have hL₁ : ContDiff 𝕜₁ n₁ L_flip1 :=
+      (ContinuousLinearMap.contDiff (𝕜 := 𝕜₁) (E := E₁' →L[𝕜₁] F₁ →L[𝕜₁] G₁)
+        (F := F₁ →L[𝕜₁] E₁' →L[𝕜₁] G₁) L_flip1).of_le le_top
+    have h2' : ContDiff 𝕜₁ n₁ (fun x => L_flip1 ((f x).comp (g x))) :=
+      ContDiff.comp hL₁ h1
+    exact (funext eq_flip).symm ▸ h2'
+  have h3 : ContDiff 𝕜₁ n₁ (fun x => (((f x).comp (g x)).flip).comp (h x)) :=
+    ContDiff.clm_comp h2 hh
+  let L_flip2 := ContinuousLinearMap.flipₗᵢ 𝕜₁ F₁' E₁' G₁
+  have eq_flip2 : ∀ x, L_flip2 ((((f x).comp (g x)).flip).comp (h x)) =
+      ((((f x).comp (g x)).flip).comp (h x)).flip := by
+    intro x
+    rfl
+  have h4 : ContDiff 𝕜₁ n₁ (fun x => ((((f x).comp (g x)).flip).comp (h x)).flip) := by
+    have hL₂ : ContDiff 𝕜₁ n₁ L_flip2 :=
+      (ContinuousLinearMap.contDiff (𝕜 := 𝕜₁) (E := F₁' →L[𝕜₁] E₁' →L[𝕜₁] G₁)
+        (F := E₁' →L[𝕜₁] F₁' →L[𝕜₁] G₁) L_flip2).of_le le_top
+    have h4' := ContDiff.comp hL₂ h3
+    exact (funext eq_flip2).symm ▸ h4'
+  exact h4
+
+variable {X₁ E₁ F₁ G₁ E₁' F₁' : Type*} [NontriviallyNormedField 𝕜₁]
+  [NormedAddCommGroup X₁] [NormedSpace 𝕜₁ X₁]
+  [NormedAddCommGroup E₁] [NormedSpace 𝕜₁ E₁] [FiniteDimensional 𝕜₁ E₁]
+  [NormedAddCommGroup F₁] [NormedSpace 𝕜₁ F₁] [FiniteDimensional 𝕜₁ F₁]
+  [NormedAddCommGroup G₁] [NormedSpace 𝕜₁ G₁] [FiniteDimensional 𝕜₁ G₁]
+  [NormedAddCommGroup E₁'] [NormedSpace 𝕜₁ E₁'] [FiniteDimensional 𝕜₁ E₁']
+  [NormedAddCommGroup F₁'] [NormedSpace 𝕜₁ F₁'] [FiniteDimensional 𝕜₁ F₁']
+  {n₁ : WithTop ℕ∞}
+
+/-- The "flip" operation on continuous bilinear maps is smooth. -/
+lemma flip_contDiff {F₁ F₂ R : Type*}
+    [NormedAddCommGroup F₁] [NormedSpace ℝ F₁]
+    [NormedAddCommGroup F₂] [NormedSpace ℝ F₂]
+    [NormedAddCommGroup R] [NormedSpace ℝ R] :
+    ContDiff ℝ ⊤ (fun f : F₁ →L[ℝ] F₂ →L[ℝ] R => ContinuousLinearMap.flip f) := by
+  let flip_clm :=
+    (ContinuousLinearMap.flipₗᵢ ℝ F₁ F₂ R).toContinuousLinearEquiv.toContinuousLinearMap
+  exact
+    @ContinuousLinearMap.contDiff ℝ _
+      (F₁ →L[ℝ] F₂ →L[ℝ] R) _ _ (F₂ →L[ℝ] F₁ →L[ℝ] R) _ _ _ flip_clm
+
+/-- Composition of a bilinear map with a linear map in the first argument is smooth. -/
+lemma comp_first_contDiff {F₁ F₂ F₃ R : Type*}
+    [NormedAddCommGroup F₁] [NormedSpace ℝ F₁]
+    [NormedAddCommGroup F₂] [NormedSpace ℝ F₂]
+    [NormedAddCommGroup F₃] [NormedSpace ℝ F₃]
+    [NormedAddCommGroup R] [NormedSpace ℝ R] :
+    ContDiff ℝ ⊤ (fun p : (F₂ →L[ℝ] F₃ →L[ℝ] R) × (F₁ →L[ℝ] F₂) =>
+      ContinuousLinearMap.comp p.1 p.2) := by
+  exact ContDiff.clm_comp contDiff_fst contDiff_snd
+
+variable {E₁_₂ : Type*} {E₂_₂ : Type*} {R₂ : Type*}
+variable [NormedAddCommGroup E₁_₂] [NormedSpace ℝ E₁_₂]
+variable [NormedAddCommGroup E₂_₂] [NormedSpace ℝ E₂_₂]
+variable [NormedAddCommGroup R₂] [NormedSpace ℝ R₂]
+
+/-- The pullback of a bilinear map by a linear map is smooth with respect to both arguments. -/
+theorem contDiff_pullbackBilinear_op :
+    ContDiff ℝ ⊤ (fun p : (E₂_₂ →L[ℝ] E₂_₂ →L[ℝ] R₂) × (E₁_₂ →L[ℝ] E₂_₂) =>
+      BilinearForm.pullback p.1 p.2) := by
+  apply contDiff_bilinearComp
+  · exact (contDiff_fst (E := (E₂_₂ →L[ℝ] E₂_₂ →L[ℝ] R₂)) (F := (E₁_₂ →L[ℝ] E₂_₂))).of_le le_top
+  · exact (contDiff_snd (E := (E₂_₂ →L[ℝ] E₂_₂ →L[ℝ] R₂)) (F := (E₁_₂ →L[ℝ] E₂_₂))).of_le le_top
+  · exact (contDiff_snd (E := (E₂_₂ →L[ℝ] E₂_₂ →L[ℝ] R₂)) (F := (E₁_₂ →L[ℝ] E₂_₂))).of_le le_top
+
+end ContinuousLinearMap
diff --git a/PhysLean/Mathematics/Geometry/Manifold/Chart/BilinearSmoothness.lean b/PhysLean/Mathematics/Geometry/Manifold/Chart/BilinearSmoothness.lean
new file mode 100644
index 000000000..8d928ce16
--- /dev/null
+++ b/PhysLean/Mathematics/Geometry/Manifold/Chart/BilinearSmoothness.lean
@@ -0,0 +1,135 @@
+/-
+Copyright (c) 2025 Matteo Cipollina. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Matteo Cipollina
+-/
+
+import Mathlib.Algebra.Lie.OfAssociative
+import Mathlib.Analysis.RCLike.Lemmas
+import Mathlib.Geometry.Manifold.IsManifold.Basic
+import PhysLean.Mathematics.Analysis.ContDiff
+
+/-!
+# Smoothness of Bilinear Forms in Chart Coordinates
+
+This file contains lemmas about the smoothness of bilinear forms in chart coordinates.
+-/
+noncomputable section
+open BilinearForm
+open Filter
+
+variable {E : Type v} {M : Type v} {n : WithTop ℕ∞}
+variable [NormedAddCommGroup E] [NormedSpace ℝ E]
+variable [TopologicalSpace M] [ChartedSpace E M] [T2Space M]
+variable {I : ModelWithCorners ℝ E E} [ModelWithCorners.Boundaryless I]
+variable [inst_mani_smooth : IsManifold I (n + 1) M] -- For C^{n+1} manifold for C^n metric
+variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
+
+noncomputable instance (x : M) : NormedAddCommGroup (TangentSpace I x) :=
+  show NormedAddCommGroup E from inferInstance
+
+noncomputable instance (x : M) : NormedSpace ℝ (TangentSpace I x) :=
+  show NormedSpace ℝ E from inferInstance
+namespace Manifold.ChartSmoothness
+
+open BilinearForm ContDiff ContinuousLinearMap
+
+variable {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X]
+variable {f_bilin : X → E →L[ℝ] E →L[ℝ] ℝ} {s_set : Set X} {x₀_pt : X}
+-- n is from the outer scope (smoothness of the metric)
+
+lemma contDiffWithinAt_eval_bilinear_apply (hf : ContDiffWithinAt ℝ n f_bilin s_set x₀_pt)
+    (v w : E) :
+    ContDiffWithinAt ℝ n (fun x => f_bilin x v w) s_set x₀_pt := by
+  let eval_vw : (E →L[ℝ] E →L[ℝ] ℝ) →L[ℝ] ℝ := BilinearForm.eval_vectors_continuousLinear v w
+  exact (eval_vw.contDiff.of_le le_top).contDiffWithinAt.comp x₀_pt hf (Set.mapsTo_univ _ _)
+
+variable[FiniteDimensional ℝ E]
+
+lemma contDiffWithinAt_bilinear_apply_iff_forall_coord :
+  (∀ v w, ContDiffWithinAt ℝ n (fun x => f_bilin x v w) s_set x₀_pt) →
+  ContDiffWithinAt ℝ n f_bilin s_set x₀_pt := by
+  intro h_comp
+  rw [ContDiff.iff_forall_coord (V := E →L[ℝ] E →L[ℝ] ℝ) (𝕜 := ℝ)]
+  intro φ
+  let b := Module.finBasis ℝ E
+  obtain ⟨e_forms, h_e_forms_def⟩ := BilinearForm.elementary_bilinear_forms_def b
+  have h_f_decomp : ∀ (x : X), f_bilin x =
+      ∑ i ∈ Finset.univ, ∑ j ∈ Finset.univ, ((f_bilin x) (b i) (b j)) • e_forms i j := by
+    intro x
+    obtain ⟨e, h_e_prop, h_decomp⟩ := BilinearForm.decomposition b (f_bilin x)
+    have e_eq_e_forms : e = e_forms := by
+      ext i j v w
+      rw [h_e_prop i j v w, h_e_forms_def i j v w]
+    rw [e_eq_e_forms] at h_decomp
+    exact h_decomp
+  have h_phi_f : ∀ x, φ (f_bilin x) =
+      ∑ i ∈ Finset.univ, ∑ j ∈ Finset.univ, ((f_bilin x) (b i) (b j)) * φ (e_forms i j) := by
+    intro x
+    rw [h_f_decomp x]
+    have h_expand : φ (∑ i ∈ Finset.univ,
+      ∑ j ∈ Finset.univ, ((f_bilin x) (b i) (b j)) • e_forms i j) =
+        ∑ i ∈ Finset.univ, φ (∑ j ∈ Finset.univ, ((f_bilin x) (b i) (b j)) • e_forms i j) :=
+        ContinuousLinearMap.map_finset_sum φ Finset.univ (fun i => ∑ j ∈ Finset.univ,
+        ((f_bilin x) (b i) (b j)) • e_forms i j)
+    rw [h_expand]
+    apply Finset.sum_congr rfl
+    intro i _
+    have h_expand_inner : φ (∑ j ∈ Finset.univ, ((f_bilin x) (b i) (b j)) • e_forms i j) =
+                          ∑ j ∈ Finset.univ, φ (((f_bilin x) (b i) (b j)) • e_forms i j) :=
+      ContinuousLinearMap.map_finset_sum φ Finset.univ (fun j =>
+        ((f_bilin x) (b i) (b j)) • e_forms i j)
+    rw [h_expand_inner]
+    apply Finset.sum_congr rfl
+    intro j _
+    rw [ContinuousLinearMap.map_smul]
+    rw [smul_eq_mul]
+    rw [← h_f_decomp]
+  have h_goal : ContDiffWithinAt ℝ n (fun x => φ (f_bilin x)) s_set x₀_pt := by
+    simp only [h_phi_f]
+    apply ContDiffWithinAt.sum
+    intro i _
+    apply ContDiffWithinAt.sum
+    intro j _
+    have h_term_smooth : ContDiffWithinAt ℝ n (fun x => f_bilin x (b i) (b j)) s_set x₀_pt :=
+      h_comp (b i) (b j)
+    have h_const : ContDiffWithinAt ℝ n (fun x => φ (e_forms i j)) s_set x₀_pt :=
+      contDiffWithinAt_const
+    exact ContDiffWithinAt.mul h_term_smooth h_const
+  exact h_goal
+
+/--
+A bilinear form `f_bilin : X → E → E → F` is continuously differentiable of order `n_level`
+at a point `x₀_pt` within a set `s_set` if and only if for all vectors `v w : E`, the function
+`x ↦ f_bilin x v w` is continuously differentiable of order `n_level` at `x₀_pt` within `s_set`.
+
+This provides an equivalence between the continuous differentiability of a bilinear map
+and the continuous differentiability of all its partial evaluations.
+-/
+theorem contDiffWithinAt_bilinear_iff {n_level : WithTop ℕ∞} :
+    (∀ (v w : E), ContDiffWithinAt ℝ n_level (fun x => f_bilin x v w) s_set x₀_pt) ↔
+    (ContDiffWithinAt ℝ n_level f_bilin s_set x₀_pt) := by
+  constructor
+  · intro h; exact contDiffWithinAt_bilinear_apply_iff_forall_coord h
+  · intro h; exact contDiffWithinAt_eval_bilinear_apply h
+
+/--
+A bilinear form `f_bilin : X → E → E → F` is continuously differentiable of order `n_level`
+on a set `s` if and only if for all vectors `v w : E`, the function `x ↦ f_bilin x v w`
+is continuously differentiable of order `n_level` on `s`.
+
+This extends `contDiffWithinAt_bilinear_iff` from a single point to an entire set.
+-/
+theorem contDiffOn_bilinear_iff {n_level : WithTop ℕ∞} (s : Set X) :
+    (∀ (v w : E), ContDiffOn ℝ n_level (fun x => f_bilin x v w) s) ↔
+    (ContDiffOn ℝ n_level f_bilin s) := by
+  simp only [ContDiffOn, contDiffWithinAt_bilinear_iff (n_level := n_level)]
+  constructor
+  · intro h x hx
+    apply contDiffWithinAt_bilinear_apply_iff_forall_coord
+    intro v w
+    exact h v w x hx
+  · intro h v w x hx
+    exact contDiffWithinAt_eval_bilinear_apply (h x hx) v w
+
+end Manifold.ChartSmoothness
diff --git a/PhysLean/Mathematics/Geometry/Manifold/Chart/CoordinateTransformations.lean b/PhysLean/Mathematics/Geometry/Manifold/Chart/CoordinateTransformations.lean
new file mode 100644
index 000000000..1aadd2ae5
--- /dev/null
+++ b/PhysLean/Mathematics/Geometry/Manifold/Chart/CoordinateTransformations.lean
@@ -0,0 +1,160 @@
+/-
+Copyright (c) 2025 Matteo Cipollina. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Matteo Cipollina
+-/
+
+import PhysLean.Mathematics.Geometry.Manifold.Chart.Smoothness
+import PhysLean.Mathematics.Geometry.Manifold.Chart.Utilities
+import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
+
+/-!
+# Coordinate Transformations and Transition Maps
+
+This file contains lemmas about coordinate transformations and transition maps between charts.
+-/
+
+namespace Manifold.Chart
+open ContDiff ContinuousLinearMap
+open PartialHomeomorph
+open Filter Manifold
+
+variable {E : Type v} {M : Type v} {n : WithTop ℕ∞}
+variable [NormedAddCommGroup E]
+variable [TopologicalSpace M]
+
+/-- The domain where the chart transition map `φ = e' ∘ e.symm` is well-defined forms
+a neighborhood of `y`. -/
+lemma chartMetric_transition_domain_in_nhds (e e' : PartialHomeomorph M E) (y : E)
+    (hy_target : y ∈ e.target) (hz_source' : e.symm y ∈ e'.source) :
+    let _ := e' ∘ e.symm
+    y ∈ e.target ∩ e.symm ⁻¹' e'.source := by
+  intro φ
+  simp only [Set.mem_inter_iff, Set.mem_preimage]
+  exact ⟨hy_target, hz_source'⟩
+
+/-- When applying the transition map `φ = e' ∘ e.symm` to a point `z` in the domain,
+the result is in `e'.target`. -/
+lemma chartMetric_transition_map_target (e e' : PartialHomeomorph M E) {z : E}
+    (hz_source' : e.symm z ∈ e'.source) :
+    let φ := e' ∘ e.symm
+    φ z ∈ e'.target := by
+  intro φ
+  exact e'.mapsTo hz_source'
+
+variable [NormedSpace ℝ E]
+variable [ChartedSpace E M]
+variable {I : ModelWithCorners ℝ E E}
+variable [inst_mani_smooth : IsManifold I (n + 1) M] -- For C^{n+1} manifold for C^n metric
+variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
+variable {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X]
+variable {f_bilin : X → E →L[ℝ] E →L[ℝ] ℝ} {s_set : Set X} {x₀_pt : X}
+
+/-- Core derivative chain rule for manifold derivatives that combines correctly with the metric. -/
+theorem manifold_derivative_chain_rule (g : PseudoRiemannianMetric E E M n I)
+    (e e' : PartialHomeomorph M E) {z : E}
+    (hz_target : z ∈ e.target) (hz_source' : e.symm z ∈ e'.source)
+    (he : e ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (he' : e' ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (hn1 : 1 ≤ n) :
+    let φ := e' ∘ e.symm
+    let ψ := e.symm.trans e'
+    let x_z := e.symm z
+    ∀ v w : E,
+      g.val x_z (mfderiv I I e.symm z v) (mfderiv I I e.symm z w) =
+        g.val x_z (mfderiv I I e'.symm (φ z) ((mfderiv I I ψ z) v))
+                 (mfderiv I I e'.symm (φ z) ((mfderiv I I ψ z) w)) := by
+  intro φ ψ x_z v w
+  have hmani : IsManifold I n M := by
+    letI := inst_mani_smooth
+    have h_le : n ≤ n + 1 := by simp [self_le_add_right]
+    exact IsManifold.of_le h_le
+  have h_chain_rule_v : mfderiv I I e'.symm (φ z) ((mfderiv I I ψ z) v) =
+                        mfderiv I I e.symm z v := by
+    let x := e.symm z
+    have h_z_eq : z = e x := Eq.symm (PartialHomeomorph.right_inv e hz_target)
+    have hx_e : x ∈ e.source := e.map_target hz_target
+    have h_fun_eq : e.symm =ᶠ[nhds z] e'.symm ∘ φ := by
+      have h_z_eq_ex : z = e x := by
+        exact h_z_eq
+      have h_lemma : e.symm =ᶠ[nhds (e x)] e'.symm ∘ ↑(e.symm.trans e') :=
+        chart_inverse_composition e e' x hx_e hz_source'
+      have h_φ_eq : φ = ↑(e.symm.trans e') := by
+        ext y
+        simp [φ, trans_apply, Function.comp_apply]
+      rw [← h_z_eq_ex, ← h_φ_eq] at h_lemma
+      exact h_lemma
+    have h_deriv_eq : mfderiv I I e.symm z = mfderiv I I (e'.symm ∘ φ) z :=
+      Filter.EventuallyEq.mfderiv_eq h_fun_eq
+    have h_chain : mfderiv I I (e'.symm ∘ φ) z v =
+                   mfderiv I I e'.symm (φ z) (mfderiv I I φ z v) := by
+      have h_comp : mfderiv I I (e'.symm ∘ φ) z =
+                    (mfderiv I I e'.symm (φ z)).comp (mfderiv I I φ z) := by
+        have hφz_target : φ z ∈ e'.target :=
+          chartMetric_phi_maps_to_target e e' hz_source'
+        have hφ_diff : MDifferentiableAt I I φ z := by
+          have h_φ_eq_ψ : φ = ψ := rfl
+          rw [h_φ_eq_ψ]
+          let x := e.symm z
+          have hx_e := e.map_target hz_target
+          have h_smooth := chart_transition_smooth e e' x hx_e hz_source' n he he'
+          simpa [h_z_eq] using h_smooth.mdifferentiableAt hn1
+        have he'_symm_diff : MDifferentiableAt I I e'.symm (φ z) := by
+          have hmani : IsManifold I n M := by
+            letI := inst_mani_smooth
+            have h : n ≤ n + 1 := by
+              simp only [self_le_add_right, φ]
+            exact IsManifold.of_le h
+          have hn1 : 1 ≤ n := by exact hn1
+          have h_symm_smooth : ContMDiffOn I I n e'.symm e'.target :=
+            contMDiffOn_symm_of_mem_maximalAtlas he'
+          exact
+            (h_symm_smooth.contMDiffAt (e'.open_target.mem_nhds hφz_target)).mdifferentiableAt hn1
+        exact mfderiv_comp_of_eq he'_symm_diff hφ_diff rfl
+      have h_apply : mfderiv I I (e'.symm ∘ φ) z v =
+                     (mfderiv I I e'.symm (φ z)).comp (mfderiv I I φ z) v := by
+        rw [h_comp]
+        exact rfl
+      rw [h_apply]
+      rfl
+    have h_φ_eq_ψ : φ = ψ := rfl
+    rw [← h_deriv_eq, h_φ_eq_ψ] at h_chain
+    exact h_chain.symm
+  have h_chain_rule_w : mfderiv I I e'.symm (φ z) ((mfderiv I I ψ z) w) =
+                      mfderiv I I e.symm z w := by
+    let x := e.symm z
+    have h_z_eq : z = e x := Eq.symm (PartialHomeomorph.right_inv e hz_target)
+    have hx_e : x ∈ e.source := e.map_target hz_target
+    have h_fun_eq : e.symm =ᶠ[nhds z] e'.symm ∘ φ := by
+      have h_z_eq_ex : z = e x := h_z_eq
+      have h_lemma := chart_inverse_composition e e' x hx_e hz_source'
+      have h_φ_eq : φ = (e.symm.trans e') := rfl
+      simpa [h_z_eq_ex, h_φ_eq] using h_lemma
+    have h_deriv_eq : mfderiv I I e.symm z = mfderiv I I (e'.symm ∘ φ) z :=
+      Filter.EventuallyEq.mfderiv_eq h_fun_eq
+    let h_comp := mfderiv_chart_transition_helper e e' x hx_e hz_source' hn1 hmani he he'
+    have h_comp_w : mfderiv I I (e'.symm ∘ φ) z w =
+                   (mfderiv I I e'.symm (φ z)).comp (mfderiv I I φ z) w := by
+      have h_comp_map : mfderiv I I (e'.symm ∘ φ) z =
+                        (mfderiv I I e'.symm (φ z)).comp (mfderiv I I φ z) := by
+        have hφz_target : φ z ∈ e'.target :=
+          chartMetric_phi_maps_to_target e e' hz_source'
+        have hφ_diff : MDifferentiableAt I I φ z := by
+          have h_φ_eq_ψ : φ = ψ := rfl
+          rw [h_φ_eq_ψ]
+          have h_smooth := chart_transition_smooth e e' x hx_e hz_source' n he he'
+          simpa [h_z_eq] using h_smooth.mdifferentiableAt hn1
+        have he'_symm_diff : MDifferentiableAt I I e'.symm (φ z) := by
+          have h_symm_smooth : ContMDiffOn I I n e'.symm e'.target :=
+            contMDiffOn_symm_of_mem_maximalAtlas he'
+          exact
+            (h_symm_smooth.contMDiffAt (e'.open_target.mem_nhds hφz_target)).mdifferentiableAt hn1
+        exact mfderiv_comp_of_eq he'_symm_diff hφ_diff rfl
+      rw [h_comp_map]
+      rfl
+    have h_φ_eq_ψ : φ = ψ := rfl
+    rw [← h_deriv_eq, h_φ_eq_ψ] at h_comp_w
+    exact h_comp_w.symm
+  rw [h_chain_rule_v, h_chain_rule_w]
+
+end Manifold.Chart
diff --git a/PhysLean/Mathematics/Geometry/Manifold/Chart/Smoothness.lean b/PhysLean/Mathematics/Geometry/Manifold/Chart/Smoothness.lean
new file mode 100644
index 000000000..e0d5f960c
--- /dev/null
+++ b/PhysLean/Mathematics/Geometry/Manifold/Chart/Smoothness.lean
@@ -0,0 +1,135 @@
+/-
+Copyright (c) 2025 Matteo Cipollina. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Matteo Cipollina
+-/
+
+import Mathlib.Geometry.Manifold.MFDeriv.Basic
+import PhysLean.Mathematics.Geometry.Manifold.PartialHomeomorph
+
+/-!
+# Smoothness of Charts and Transition Maps
+
+This file contains lemmas about the smoothness and differentiability of charts and transition maps.
+-/
+
+namespace Manifold.Chart
+
+variable {E : Type v} {M : Type v} {n : WithTop ℕ∞}
+variable [NormedAddCommGroup E]
+variable [TopologicalSpace M]
+
+open PartialHomeomorph ContinuousLinearMap PartialEquiv
+
+/-- Coordinate transformation identity: e' x = φ(e x) where φ is the transition map. -/
+lemma chart_coordinate_transform (e e' : PartialHomeomorph M E) (x : M)
+    (hx_e : x ∈ e.source) :
+    let φ := e.symm.trans e';
+    let y := e x;
+    e' x = φ y := (apply_transition_map_eq_chart_apply e e' x hx_e).symm
+
+/-- Chart inverse composition identity: e.symm = e'.symm ∘ φ in a neighborhood of e x,
+    where φ is the transition map. -/
+lemma chart_inverse_composition (e e' : PartialHomeomorph M E) (x : M)
+    (hx_e : x ∈ e.source) (hx_e' : x ∈ e'.source) :
+    let φ := e.symm.trans e';
+    let y := e x;
+    e.symm =ᶠ[nhds y] e'.symm ∘ φ := by
+  intro φ y
+  have hy_dom_φ : y ∈ φ.source :=
+    transition_map_source_contains_image_point e e' x hx_e hx_e'
+  filter_upwards [φ.open_source.mem_nhds hy_dom_φ] with z hz
+  exact apply_symm_eq_apply_symm_comp_transition e e' z hz
+
+variable [NormedSpace ℝ E]
+variable [ChartedSpace E M]
+variable {I : ModelWithCorners ℝ E E}
+
+open PartialHomeomorph ContinuousLinearMap PartialEquiv
+
+/-- Transition maps between charts in the maximal atlas are smooth. -/
+lemma chart_transition_smooth (e e' : PartialHomeomorph M E) (x : M)
+    (hx_e : x ∈ e.source) (hx_e' : x ∈ e'.source)
+    (n : WithTop ℕ∞)
+    (he : e ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (he' : e' ∈ (contDiffGroupoid n I).maximalAtlas M) :
+    let φ := e.symm.trans e';
+    let y := e x;
+    ContMDiffAt I I n φ y := by
+  intro φ y
+  have hy_dom_φ : y ∈ φ.source :=
+    transition_map_source_contains_image_point e e' x hx_e hx_e'
+  have h_φ_groupoid : φ ∈ contDiffGroupoid n I :=
+    (contDiffGroupoid n I).compatible_of_mem_maximalAtlas he he'
+  have h_φ_smooth : ContMDiffOn I I n φ φ.source :=
+    contMDiffOn_of_mem_contDiffGroupoid h_φ_groupoid
+  exact ContMDiffOn.contMDiffAt h_φ_smooth (φ.open_source.mem_nhds hy_dom_φ)
+
+variable {E : Type v} {M : Type v} {n : WithTop ℕ∞}
+variable [NormedAddCommGroup E] [NormedSpace ℝ E] --[FiniteDimensional ℝ E]
+variable [TopologicalSpace M] [ChartedSpace E M] --[T2Space M]
+variable {I : ModelWithCorners ℝ E E} --[ModelWithCorners.Boundaryless I]
+
+/-- Charts in the maximal atlas are differentiable on their source. -/
+lemma chart_differentiable_on_source
+    (e : PartialHomeomorph M E) (hmani : IsManifold I n M)
+    (he : e ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (hn1 : 1 ≤ n) :
+    MDifferentiableOn I I e e.source := by
+  intro x hx
+  have h_smooth : ContMDiffAt I I n e x :=
+    (contMDiffOn_of_mem_maximalAtlas he).contMDiffAt (e.open_source.mem_nhds hx)
+  exact (h_smooth.mdifferentiableWithinAt hn1).mono (by exact fun ⦃a⦄ a => trivial)
+
+lemma chart_symm_differentiable_on_target
+    (e : PartialHomeomorph M E) (hmani : IsManifold I n M)
+    (he : e ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (hn1 : 1 ≤ n) :
+    MDifferentiableOn I I e.symm e.target := by
+  intro y hy
+  have h_smooth : ContMDiffAt I I n e.symm y :=
+    (contMDiffOn_symm_of_mem_maximalAtlas he).contMDiffAt (e.open_target.mem_nhds hy)
+  let h_md := h_smooth.mdifferentiableWithinAt hn1
+  exact h_md.mono (Set.subset_univ _)
+
+/-- The manifold derivative of a chart transition map can be computed from charts. -/
+lemma mfderiv_chart_transition_helper (e e' : PartialHomeomorph M E) (x : M)
+    (hx_e : x ∈ e.source) (hx_e' : x ∈ e'.source)
+    (hn1 : 1 ≤ n) (hmani : IsManifold I n M)
+    (he : e ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (he' : e' ∈ (contDiffGroupoid n I).maximalAtlas M) :
+    let φ := e.symm.trans e';
+    let y := e x;
+    mfderiv I I (e'.symm ∘ φ) y = (mfderiv I I e'.symm (φ y)).comp (mfderiv I I φ y) := by
+  intro φ y
+  have h_φ_y_eq : φ y = e' x := (chart_coordinate_transform e e' x hx_e).symm
+  have hφ_diff : ContMDiffAt I I n φ y :=
+    chart_transition_smooth e e' x hx_e hx_e' n he he'
+  have hφ_mdiff : MDifferentiableAt I I φ y := hφ_diff.mdifferentiableAt hn1
+  have he'_symm_diff : MDifferentiableAt I I e'.symm (e' x) :=
+    ((chart_symm_differentiable_on_target e' hmani he' hn1) (e' x)
+      (e'.mapsTo hx_e')).mdifferentiableAt
+      (e'.open_target.mem_nhds (e'.mapsTo hx_e'))
+  exact mfderiv_comp_of_eq he'_symm_diff hφ_mdiff h_φ_y_eq
+
+/-- Chain rule for composition of chart inverses with transition maps. -/
+theorem mfderiv_chart_transition (e e' : PartialHomeomorph M E) (x : M)
+    (hx_e : x ∈ e.source) (hx_e' : x ∈ e'.source)
+    (hn1 : 1 ≤ n) (hmani : IsManifold I n M)
+    (he : e ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (he' : e' ∈ (contDiffGroupoid n I).maximalAtlas M) :
+    let φ := e.symm.trans e';
+    let y := e x;
+    mfderiv I I e.symm y = (mfderiv I I e'.symm (e' x)).comp (mfderiv I I φ y) := by
+  intro φ y
+  have h_fun_eq : e.symm =ᶠ[nhds y] e'.symm ∘ φ :=
+    chart_inverse_composition e e' x hx_e hx_e'
+  have h_φ_y_eq : φ y = e' x :=
+    (chart_coordinate_transform e e' x hx_e).symm
+  have h_deriv_eq : mfderiv I I e.symm y = mfderiv I I (e'.symm ∘ φ) y :=
+    Filter.EventuallyEq.mfderiv_eq h_fun_eq
+  rw [h_deriv_eq]
+  have h_comp := mfderiv_chart_transition_helper e e' x hx_e hx_e' hn1 hmani he he'
+  rw [h_comp, h_φ_y_eq]
+
+end Manifold.Chart
diff --git a/PhysLean/Mathematics/Geometry/Manifold/Chart/Utilities.lean b/PhysLean/Mathematics/Geometry/Manifold/Chart/Utilities.lean
new file mode 100644
index 000000000..fc1600faa
--- /dev/null
+++ b/PhysLean/Mathematics/Geometry/Manifold/Chart/Utilities.lean
@@ -0,0 +1,59 @@
+/-
+Copyright (c) 2025 Matteo Cipollina. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Matteo Cipollina
+-/
+
+import Mathlib.Analysis.Normed.Group.Basic
+import Mathlib.Topology.PartialHomeomorph
+/-!
+# Utilities for Charts and Transition Maps
+
+This file contains general utility lemmas for charts and transition maps.
+-/
+
+namespace Manifold.Chart
+variable {E : Type v} {M : Type v} {n : WithTop ℕ∞}
+variable [NormedAddCommGroup E]
+variable [TopologicalSpace M]
+
+/-- The domain where the chart transition is well-defined forms a neighborhood of y. -/
+lemma chartMetric_domain_in_nhds (e e' : PartialHomeomorph M E) (y : E)
+    (hy : y ∈ e.target) (hx_source' : e.symm y ∈ e'.source) :
+    let s := e.target ∩ e.symm ⁻¹' e'.source
+    s ∈ nhds y := by
+  intro s
+  apply Filter.inter_mem
+  · exact e.open_target.mem_nhds hy
+  · exact (e.continuousAt_symm hy).preimage_mem_nhds (e'.open_source.mem_nhds hx_source')
+
+/-- When applying φ = e' ∘ e.symm to a point z in the domain s, the result is in e'.target -/
+lemma chartMetric_phi_in_target (e e' : PartialHomeomorph M E) {z : E}
+    (hz_source' : e.symm z ∈ e'.source) :
+    let φ := e'.toPartialEquiv ∘ e.symm
+    φ z ∈ e'.target := by
+  intro φ
+  unfold φ
+  exact e'.mapsTo hz_source'
+
+/-- When applying φ = e' ∘ e.symm to a point z, the result is in e'.target -/
+lemma chartMetric_phi_maps_to_target (e e' : PartialHomeomorph M E) {z : E}
+    (hz_source' : e.symm z ∈ e'.source) :
+    let φ := e' ∘ e.symm
+    φ z ∈ e'.target := by
+  intro φ
+  unfold φ
+  exact e'.mapsTo hz_source'
+
+/-- The composition of chart maps preserves points: e'.symm (e' (e.symm z)) = e.symm z -/
+lemma chart_map_composition_identity (e e' : PartialHomeomorph M E) {z : E}
+    (hz_source' : e.symm z ∈ e'.source) :
+    let φ := e' ∘ e.symm
+    let x_z := e.symm z
+    e'.symm (φ z) = x_z := by
+  intro φ x_z
+  unfold φ
+  simp only [Function.comp_apply, PartialHomeomorph.coe_coe]
+  rw [e'.left_inv hz_source']
+
+end Manifold.Chart
diff --git a/PhysLean/Mathematics/Geometry/Manifold/PartialHomeomorph.lean b/PhysLean/Mathematics/Geometry/Manifold/PartialHomeomorph.lean
new file mode 100644
index 000000000..413363084
--- /dev/null
+++ b/PhysLean/Mathematics/Geometry/Manifold/PartialHomeomorph.lean
@@ -0,0 +1,209 @@
+/-
+Copyright (c) 2025 Matteo Cipollina. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Matteo Cipollina
+-/
+
+import Mathlib.Geometry.Manifold.ContMDiff.Atlas
+import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
+
+/-!
+# PartialHomeomorph Utilities for Manifolds
+
+This file contains lemmas for `PartialHomeomorph` specifically relevant to manifolds.
+-/
+
+namespace PartialHomeomorph
+
+open Set ModelWithCorners PartialEquiv
+
+variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable {H : Type*} [NormedAddCommGroup H] [NormedSpace 𝕜 H] [TopologicalSpace H]
+variable {M : Type*} [TopologicalSpace M] [ChartedSpace H M]
+variable {I : ModelWithCorners 𝕜 E H} {n : WithTop ℕ∞}
+
+/-- The toPartialEquiv of the symm of a partial homeomorphism equals the symm of its
+toPartialEquiv. -/
+@[simp, mfld_simps]
+theorem toPartialEquiv_symm {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+    (e : PartialHomeomorph X Y) :
+    e.symm.toPartialEquiv = e.toPartialEquiv.symm :=
+  rfl
+
+/-- The function component of the symm of a partial homeomorphism equals its invFun. -/
+theorem coe_symm_eq_invFun {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+    (e : PartialHomeomorph X Y) : ⇑e.symm = e.invFun :=
+  rfl
+
+omit [NormedAddCommGroup H] [NormedSpace 𝕜 H] in
+/-- If a partial homeomorphism belongs to the `contDiffGroupoid n I`, then the forward map is
+`C^n` on its source. This extracts the smoothness property directly from groupoid membership. -/
+theorem contMDiffOn_of_mem_groupoid {e : PartialHomeomorph H H}
+    (he : e ∈ contDiffGroupoid n I) : ContMDiffOn I I n e e.source :=
+  contMDiffOn_of_mem_contDiffGroupoid he
+
+omit [NormedAddCommGroup H] [NormedSpace 𝕜 H] in
+/-- The coordinate change map `φ = e' ∘ e.symm` between two charts `e` and `e'` is `C^n` smooth
+    at a point `y = e x` if `x` is in the intersection of the sources of `e` and `e'`. -/
+lemma contMDiffAt_coord_change
+    {e e' : PartialHomeomorph M H} {x : M}
+    (hx_e_source : x ∈ e.source) (hx_e'_source : x ∈ e'.source)
+    (he : e ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (he' : e' ∈ (contDiffGroupoid n I).maximalAtlas M) :
+    ContMDiffAt I I n (e' ∘ e.symm) (e x) := by
+  let y := e x
+  have hy_target : y ∈ e.target := e.mapsTo hx_e_source
+  let transition_domain := e.target ∩ e.symm ⁻¹' e'.source
+  have hy_domain : y ∈ transition_domain := ⟨hy_target, by
+    rw [@Set.mem_preimage];
+    rw [e.left_inv hx_e_source];
+    exact hx_e'_source⟩
+  have open_domain : IsOpen transition_domain := e.isOpen_inter_preimage_symm e'.open_source
+  have h_compatible := StructureGroupoid.compatible_of_mem_maximalAtlas he he'
+  have h_transition_smooth : ContMDiffOn I I n (e' ∘ e.symm) transition_domain :=
+    contMDiffOn_of_mem_groupoid h_compatible
+  exact h_transition_smooth.contMDiffAt (open_domain.mem_nhds hy_domain)
+
+omit [NormedAddCommGroup H] [NormedSpace 𝕜 H] in
+/-- The coordinate change map between atlas charts is smooth in the manifold sense. -/
+lemma contMDiffAt_atlas_coord_change [IsManifold I n M]
+    {e e' : PartialHomeomorph M H}
+    (he : e ∈ atlas H M) (he' : e' ∈ atlas H M) {x : M}
+    (hx_e_source : x ∈ e.source) (hx_e'_source : x ∈ e'.source) :
+    ContMDiffAt I I n (e' ∘ e.symm) (e x) := by
+  let transition_domain := e.target ∩ e.symm ⁻¹' e'.source
+  have open_domain : IsOpen transition_domain := e.isOpen_inter_preimage_symm e'.open_source
+  have hy_target : e x ∈ e.target := e.mapsTo hx_e_source
+  have hy_domain : e x ∈ transition_domain := ⟨hy_target, by
+    show e.symm (e x) ∈ e'.source
+    rw [e.left_inv hx_e_source]
+    exact hx_e'_source⟩
+  have h_compatible : e.symm.trans e' ∈ contDiffGroupoid n I := HasGroupoid.compatible he he'
+  have h_transition_smooth : ContMDiffOn I I n (e' ∘ e.symm) transition_domain :=
+    contMDiffOn_of_mem_groupoid h_compatible
+  exact h_transition_smooth.contMDiffAt (open_domain.mem_nhds hy_domain)
+
+/-- The coordinate change map `φ = e' ∘ e.symm` between two charts `e` and `e'` taken from the
+    atlas is `C^n` smooth at a point `y = e x` if `x` is in the intersection of their sources. -/
+lemma contDiffAt_atlas_coord_change
+    {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
+    {n : WithTop ℕ∞} [ChartedSpace E M]
+    [IsManifold (modelWithCornersSelf 𝕜 E) n M]
+    {e e' : PartialHomeomorph M E}
+    (he : e ∈ atlas E M) (he' : e' ∈ atlas E M) {x : M}
+    (hx_e_source : x ∈ e.source) (hx_e'_source : x ∈ e'.source) :
+    let y := e x
+    ContDiffAt 𝕜 n (e' ∘ e.symm) y := by
+  intro y
+  have hy_target : y ∈ e.target := e.mapsTo hx_e_source
+  let transition_domain := e.target ∩ e.symm ⁻¹' e'.source
+  have hy_domain : y ∈ transition_domain := by
+    refine ⟨hy_target, ?_⟩
+    simp only [Set.mem_preimage]
+    rw [e.left_inv hx_e_source]
+    exact hx_e'_source
+  have h_compatible : e.symm.trans e' ∈ contDiffGroupoid n (modelWithCornersSelf 𝕜 E) :=
+    HasGroupoid.compatible he he'
+  have h_transition_mfd_smooth : ContMDiffOn (modelWithCornersSelf 𝕜 E)
+      (modelWithCornersSelf 𝕜 E) n (e' ∘ e.symm) transition_domain :=
+    contMDiffOn_of_mem_contDiffGroupoid h_compatible
+  have h_transition_smooth : ContDiffOn 𝕜 n (e' ∘ e.symm) transition_domain :=
+    (contMDiffOn_iff_contDiffOn).mp h_transition_mfd_smooth
+  have open_domain : IsOpen transition_domain :=
+    ContinuousOn.isOpen_inter_preimage e.continuousOn_invFun e.open_target e'.open_source
+  exact h_transition_smooth.contDiffAt (IsOpen.mem_nhds open_domain hy_domain)
+
+lemma extChartAt_eq_extend_chartAt {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕜 E']
+    (I' : ModelWithCorners 𝕜 E' E') (x_m : M) [ChartedSpace E' M] :
+    extChartAt I' x_m = (chartAt E' x_m).extend I' :=
+  rfl
+
+variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+variable [K_bl : I.Boundaryless] -- Changed K to I
+
+omit [NormedSpace 𝕜 E] in
+/-- Domain of the transition map `e.symm.trans e'` contains `e x`. -/
+lemma transition_map_source_contains_image_point {E_chart : Type*} [TopologicalSpace E_chart]
+    (e e' : PartialHomeomorph M E_chart) (x : M)
+    (hx_e_source : x ∈ e.source) (hx_e'_source : x ∈ e'.source) :
+    (e x) ∈ (e.symm.toPartialEquiv.trans e'.toPartialEquiv).source := by
+  rw [@PartialEquiv.trans_source'']
+  simp only [symm_toPartialEquiv, PartialEquiv.symm_symm, toFun_eq_coe, PartialEquiv.symm_target,
+    Set.mem_image, Set.mem_inter_iff]
+  rw [← @symm_image_target_eq_source]
+  rw [@symm_image_target_eq_source]
+  apply Exists.intro
+  · apply And.intro
+    on_goal 2 => {rfl
+    }
+    · simp_all only [and_self]
+
+variable {f_map : H → H} {y_pt : H} -- Renamed f, y to avoid clash
+
+omit [NormedAddCommGroup H] [NormedSpace 𝕜 H] K_bl in
+lemma contDiffAt_chart_expression_of_contMDiffAt [I.Boundaryless]
+    (h_contMDiff : ContMDiffAt I I n f_map y_pt) :
+    ContDiffAt 𝕜 n (I ∘ f_map ∘ I.symm) (I y_pt) := by
+  rw [@contMDiffAt_iff] at h_contMDiff
+  obtain ⟨_, h_diff⟩ := h_contMDiff
+  have h_fun_eq : (extChartAt I (f_map y_pt)) ∘ f_map ∘ (extChartAt I y_pt).symm =
+      I ∘ f_map ∘ I.symm := by
+    simp only [extChartAt, extend, refl_partialEquiv, PartialEquiv.refl_source,
+      singletonChartedSpace_chartAt_eq, PartialEquiv.refl_trans, toPartialEquiv_coe,
+      toPartialEquiv_coe_symm]
+  have h_point_eq : (extChartAt I y_pt) y_pt = I y_pt := by
+    simp only [extChartAt, extend, refl_partialEquiv, PartialEquiv.refl_source,
+      singletonChartedSpace_chartAt_eq, PartialEquiv.refl_trans, toPartialEquiv_coe]
+  have h_diff' : ContDiffWithinAt 𝕜 n (I ∘ f_map ∘ I.symm) (Set.range I) (I y_pt) := by
+    rw [← h_fun_eq, ← h_point_eq]
+    exact h_diff
+  apply ContDiffWithinAt.contDiffAt h_diff'
+  have : I y_pt ∈ interior (Set.range I) := by
+    have h_range : Set.range I = Set.univ := I.range_eq_univ
+    simp [h_range, interior_univ]
+  exact mem_interior_iff_mem_nhds.mp this
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+variable {I : ModelWithCorners ℝ E E} [I.Boundaryless]
+variable {n : WithTop ℕ∞}
+
+lemma contDiff_of_boundaryless_model_on_self :
+    ContDiff ℝ n (modelWithCornersSelf ℝ E).toFun := by
+  let I := modelWithCornersSelf ℝ E
+  have h_mem_groupoid : PartialHomeomorph.refl E ∈ contDiffGroupoid n I :=
+    StructureGroupoid.id_mem (contDiffGroupoid n I)
+  have h_contMDiffOn : ContMDiffOn I I n (PartialHomeomorph.refl E) Set.univ :=
+    contMDiffOn_of_mem_contDiffGroupoid h_mem_groupoid
+  rw [contMDiffOn_iff_contDiffOn] at h_contMDiffOn
+  simpa [contDiffOn_univ] using h_contMDiffOn
+
+lemma contDiff_symm_of_boundaryless_model_on_self :
+    ContDiff ℝ n (modelWithCornersSelf ℝ E).symm.toFun := by
+  let I := modelWithCornersSelf ℝ E
+  have h_mem_groupoid : PartialHomeomorph.refl E ∈ contDiffGroupoid n I :=
+    StructureGroupoid.id_mem (contDiffGroupoid n I)
+  have h_mdf := contMDiffOn_of_mem_contDiffGroupoid h_mem_groupoid
+  exact contDiffOn_univ.1 (contMDiffOn_iff_contDiffOn.1 h_mdf)
+
+omit [NormedSpace ℝ E]  in
+/-- Functional equality e.symm = e'.symm ∘ φ-/
+lemma apply_symm_eq_apply_symm_comp_transition (e e' : PartialHomeomorph M E) :
+    let φ_pe := e.symm.toPartialEquiv.trans e'.toPartialEquiv;
+    ∀ z ∈ φ_pe.source, e.symm z = (e'.symm ∘ ⇑φ_pe) z := by
+  intro φ_pe z hz_in_φ_source
+  rw [PartialEquiv.trans_source, Set.mem_inter_iff, Set.mem_preimage] at hz_in_φ_source
+  rcases hz_in_φ_source with ⟨_, hz_esymm_z_in_eprime_source⟩
+  simp only [Function.comp_apply, PartialEquiv.trans_apply,
+             PartialHomeomorph.coe_coe]
+  exact Eq.symm (e'.left_inv' hz_esymm_z_in_eprime_source)
+
+omit [NormedSpace ℝ E]  in
+/-- Image of y under φ-/
+lemma apply_transition_map_eq_chart_apply (e e' : PartialHomeomorph M E) (x : M)
+    (hx_e_source : x ∈ e.source) :
+    (e.symm.toPartialEquiv.trans e'.toPartialEquiv) (e x) = e' x := by
+  simp only [PartialEquiv.trans_apply, PartialHomeomorph.coe_coe,
+             e.left_inv hx_e_source]
+
+end PartialHomeomorph
diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean
new file mode 100644
index 000000000..66c1c43bb
--- /dev/null
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean
@@ -0,0 +1,142 @@
+/-
+Copyright (c) 2025 Matteo Cipollina. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Matteo Cipollina
+-/
+
+import Mathlib.Algebra.Lie.OfAssociative
+import PhysLean.Mathematics.Geometry.Manifold.Chart.Smoothness
+import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
+import PhysLean.Mathematics.LinearAlgebra.BilinearForm
+
+/-!
+# Pseudo-Riemannian Metric in Chart Coordinates
+
+This file defines `chartMetric`, which expresses a pseudo-Riemannian metric in local chart
+coordinates, and proves its transformation properties under coordinate changes.
+-/
+
+namespace PseudoRiemannianMetric
+
+noncomputable section
+open BilinearForm PartialHomeomorph ContinuousLinearMap PartialEquiv ContDiff Manifold.Chart
+
+open Filter
+
+variable {E : Type v} {M : Type v} {n : WithTop ℕ∞}
+variable [NormedAddCommGroup E] [NormedSpace ℝ E] --[FiniteDimensional ℝ E]
+variable [TopologicalSpace M] [ChartedSpace E M] --[T2Space M]
+variable {I : ModelWithCorners ℝ E E} --[ModelWithCorners.Boundaryless I]
+variable [inst_mani_smooth : IsManifold I (n + 1) M] -- For C^{n+1} manifold for C^n metric
+variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
+
+noncomputable instance (x : M) : NormedAddCommGroup (TangentSpace I x) :=
+  show NormedAddCommGroup E from inferInstance
+
+noncomputable instance (x : M) : NormedSpace ℝ (TangentSpace I x) :=
+  show NormedSpace ℝ E from inferInstance
+
+/-- Given a pseudo-Riemannian metric `g` and a partial homeomorphism `e`, computes the metric
+in the chart coordinates. At a point `y` in the target of `e`, the result is the pullback of
+the metric at `e.symm y` by the derivative of `e.symm` at `y`. -/
+def chartMetric (g : PseudoRiemannianMetric E E M n I) (e : PartialHomeomorph M E) (y : E) :
+    E →L[ℝ] E →L[ℝ] ℝ :=
+  letI := Classical.propDecidable
+  dite (y ∈ e.target)
+    (fun _ : y ∈ e.target =>
+      let x := e.symm y
+      letI : FiniteDimensional ℝ (TangentSpace I x) := inferInstance
+      let Df : E →L[ℝ] TangentSpace I x := mfderiv I I e.symm y
+      BilinearForm.pullback (g.val x) Df)
+    (fun _ => (0 : E →L[ℝ] E →L[ℝ] ℝ))
+
+@[simp]
+lemma chartMetric_apply_of_mem (g : PseudoRiemannianMetric E E M n I) (e : PartialHomeomorph M E)
+    {y : E} (hy : y ∈ e.target) (v w : E) :
+    chartMetric g e y v w = g.val (e.symm y) (mfderiv I I e.symm y v) (mfderiv I I e.symm y w) := by
+  letI := Classical.propDecidable
+  rw [chartMetric, dite_eq_ite, if_pos hy]
+  simp only [BilinearForm.pullback, ContinuousLinearMap.bilinearComp_apply]
+  exact rfl
+
+lemma chartMetric_apply_of_not_mem
+    (g : PseudoRiemannianMetric E E M n I)
+    (e : PartialHomeomorph M E)
+    {y : E} (hy : y ∉ e.target) (v w : E) :
+    chartMetric g e y v w = 0 := by
+  simp only [chartMetric, hy, ↓reduceDIte, ContinuousLinearMap.zero_apply]
+
+/-- The value of a metric in chart coordinates at corresponding points. -/
+lemma chartMetric_at_corresponding_points (g : PseudoRiemannianMetric E E M n I)
+    (e e' : PartialHomeomorph M E) (x : M)
+    (hx_e : x ∈ e.source) (hx_e' : x ∈ e'.source)
+    (hmani : IsManifold I n M)
+    (hn1 : 1 ≤ n)
+    (he : e ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (he' : e' ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (v w : E) :
+    let y := e x
+    let y' := e' x
+    g.val x (mfderiv I I e.symm y v) (mfderiv I I e.symm y w) =
+    g.val x (mfderiv I I e'.symm y' (mfderiv I I (e.symm.trans e') y v))
+            (mfderiv I I e'.symm y' (mfderiv I I (e.symm.trans e') y w)) := by
+  intro y y'
+  have h_chain := mfderiv_chart_transition e e' x hx_e hx_e'
+                    hn1 hmani he he'
+  rw [h_chain]
+  congr
+
+/-- The relationship between chart metrics under coordinate change. -/
+lemma chartMetric_bilinear_pullback (g : PseudoRiemannianMetric E E M n I)
+    (e e' : PartialHomeomorph M E) (x : M)
+    (hx_e : x ∈ e.source) (hx_e' : x ∈ e'.source)
+    (hmani : IsManifold I n M)
+    (hn1 : 1 ≤ n)
+    (he : e ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (he' : e' ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (v w : E) :
+    let y := e x
+    let y' := e' x
+    let φ := e.symm.trans e'
+    chartMetric g e y v w =
+    (BilinearForm.pullback (chartMetric g e' y') (mfderiv I I φ y)) v w := by
+  intro y y' φ
+  have hy_target : y ∈ e.target := e.mapsTo hx_e
+  have hy'_target : y' ∈ e'.target := e'.mapsTo hx_e'
+  have h_x_e : e.symm y = x := e.left_inv hx_e
+  have h_x_e' : e'.symm y' = x := e'.left_inv hx_e'
+  simp only [chartMetric_apply_of_mem g e hy_target,
+    chartMetric_apply_of_mem g e' hy'_target,
+    BilinearForm.pullback, ContinuousLinearMap.bilinearComp_apply]
+  rw [h_x_e]
+  rw [h_x_e']
+  have h_phi_def : φ = e.symm.trans e' := rfl
+  rw [h_phi_def]
+  exact chartMetric_at_corresponding_points g e e' x hx_e hx_e' hmani hn1 he he' v w
+
+/--
+The metric tensor transformation law under chart changes.
+
+This theorem states that the representation of a pseudo-Riemannian metric in different charts
+are related by the pullback of the coordinate transformation between these charts.
+Specifically, if `e` and `e'` are two charts containing a point `x_pt`, then the metric
+expressed in chart `e` equals the pullback of the metric expressed in chart `e'`
+by the differential of the transition map from `e` to `e'`.
+-/
+theorem chartMetric_coord_change (g : PseudoRiemannianMetric E E M n I)
+    (e e' : PartialHomeomorph M E)
+    (x_pt : M)
+    (hx_e_source : x_pt ∈ e.source) (hx_e'_source : x_pt ∈ e'.source)
+    (hn1 : 1 ≤ n)
+    (hmani_chart : IsManifold I n M)
+    (he_atlas : e ∈ (contDiffGroupoid n I).maximalAtlas M)
+    (he'_atlas : e' ∈ (contDiffGroupoid n I).maximalAtlas M) :
+    chartMetric g e (e x_pt) =
+      BilinearForm.pullback (chartMetric g e' (e' x_pt))
+        (mfderiv I I (e.symm.trans e') (e x_pt)) := by
+  let φ := e.symm.trans e'
+  let y := e x_pt
+  let y' := e' x_pt
+  ext v w
+  exact chartMetric_bilinear_pullback g e e' x_pt hx_e_source hx_e'_source
+                                     hmani_chart hn1 he_atlas he'_atlas v w
diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index ddf5ec1cb..f5661003f 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -3,12 +3,14 @@ Copyright (c) 2025 Matteo Cipollina. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Matteo Cipollina
 -/
+
 import Mathlib.Analysis.InnerProductSpace.Basic
 import Mathlib.Analysis.RCLike.Lemmas
 import Mathlib.Geometry.Manifold.MFDeriv.Defs
 import Mathlib.LinearAlgebra.BilinearForm.Properties
 import Mathlib.LinearAlgebra.QuadraticForm.Real
 import Mathlib.Topology.LocallyConstant.Basic
+
 /-!
 # Pseudo-Riemannian Metrics on Smooth Manifolds
 
@@ -164,9 +166,9 @@ private def pseudoRiemannianMetricValToQuadraticForm
               ring⟩
 
 /-- A pseudo-Riemannian metric of smoothness class `C^n` on a manifold `M` modelled on `(E, H)`
-    with model `I`. This structure defines a smoothly varying, non-degenerate, symmetric,
-    continuous bilinear form `gₓ` of constant negative dimension on the tangent space `TₓM`
-    at each point `x`. Requires `M` to be `C^{n+1}` smooth.-/
+with model `I`. This structure defines a smoothly varying, non-degenerate, symmetric,
+continuous bilinear form `gₓ` of constant negative dimension on the tangent space `TₓM`
+at each point `x`. Requires `M` to be `C^{n+1}` smooth.-/
 @[ext]
 structure PseudoRiemannianMetric
     (E : Type v) (H : Type w) (M : Type w) (n : WithTop ℕ∞)
diff --git a/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean b/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean
new file mode 100644
index 000000000..1ed42aeaa
--- /dev/null
+++ b/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean
@@ -0,0 +1,185 @@
+/-
+Copyright (c) 2025 Matteo Cipollina. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Matteo Cipollina
+-/
+
+import Mathlib.Analysis.InnerProductSpace.Basic
+import Mathlib.Analysis.Normed.Module.FiniteDimension
+import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
+import Mathlib.Topology.Algebra.Module.ModuleTopology
+import Mathlib.Topology.Metrizable.CompletelyMetrizable
+
+/-!
+# Bilinear Form Utilities
+
+This file contains general lemmas and definitions related to bilinear forms,
+particularly in the context of finite-dimensional normed spaces.
+-/
+
+namespace BilinearForm
+
+noncomputable section
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+/-- Given a bilinear map `B` and a linear map `A`, constructs the pullback of `B` by `A`,
+which is the bilinear map `(v, w) ↦ B (A v) (A w)`. -/
+abbrev pullback {F₁ F₂ R : Type*} [NormedAddCommGroup F₁] [NormedSpace ℝ F₁]
+    [NormedAddCommGroup F₂] [NormedSpace ℝ F₂] [NormedAddCommGroup R] [NormedSpace ℝ R]
+    (B : F₂ →L[ℝ] F₂ →L[ℝ] R) (A : F₁ →L[ℝ] F₂) : F₁ →L[ℝ] F₁ →L[ℝ] R :=
+  ContinuousLinearMap.bilinearComp B A A
+
+/-- For a finite-dimensional space E with basis b, constructs elementary bilinear forms e_ij
+    such that e_ij(v, w) = (b.coord i)(v) * (b.coord j)(w). -/
+lemma elementary_bilinear_forms_def [FiniteDimensional ℝ E]
+    (b : Basis (Fin (Module.finrank ℝ E)) ℝ E) :
+    ∃ (e : (Fin (Module.finrank ℝ E)) → (Fin (Module.finrank ℝ E)) → (E →L[ℝ] E →L[ℝ] ℝ)),
+    ∀ (i j : Fin (Module.finrank ℝ E)) (v w : E),
+    e i j v w = (b.coord i) v * (b.coord j) w := by
+  let n := Module.finrank ℝ E
+  let idx := Fin n
+  let b_dual (i : idx) := b.coord i
+  let e (i j : idx) : E →L[ℝ] E →L[ℝ] ℝ :=
+    let fi : E →L[ℝ] ℝ := LinearMap.toContinuousLinearMap (b_dual i)
+    let fj : E →L[ℝ] ℝ := LinearMap.toContinuousLinearMap (b_dual j)
+    let mul_map : ℝ →L[ℝ] ℝ →L[ℝ] ℝ := ContinuousLinearMap.mul ℝ ℝ
+    ContinuousLinearMap.bilinearComp mul_map fi fj
+  have h_prop : ∀ (i j : idx) (v w : E), e i j v w = (b.coord i) v * (b.coord j) w := by
+    intro i j v w
+    simp only [Basis.coord_apply]
+    rfl
+  exact ⟨e, h_prop⟩
+
+/-- Every vector in a finite-dimensional space can be written as a sum
+    of basis vectors with coordinates given by the dual basis. -/
+lemma vector_basis_expansion
+    (b : Basis (Fin (Module.finrank ℝ E)) ℝ E) (v : E) :
+    v = ∑ i ∈ Finset.univ, (b.coord i) v • b i := by
+  rw [← b.sum_repr v]
+  congr
+  ext i
+  simp only [b.coord_apply, Finsupp.single_apply]
+  rw [@Basis.repr_sum_self]
+
+lemma sum_smul_left (L : E →L[ℝ] E →L[ℝ] ℝ) {ι : Type*}
+    (s : Finset ι) (c : ι → ℝ) (v : ι → E) (w : E) :
+    L (∑ i ∈ s, c i • v i) w = ∑ i ∈ s, c i • L (v i) w := by
+  simp_rw [map_sum, ContinuousLinearMap.map_smul]
+  exact ContinuousLinearMap.sum_apply s (fun d => c d • L (v d)) w
+
+lemma sum_mul_left (L : E →L[ℝ] E →L[ℝ] ℝ) {ι : Type*}
+    (s : Finset ι) (c : ι → ℝ) (v : ι → E) (w : E) :
+    L (∑ i ∈ s, c i • v i) w = ∑ i ∈ s, c i * L (v i) w := by
+  rw [sum_smul_left L s c v w]
+  simp_rw [smul_eq_mul]
+
+lemma sum_smul_right (L : E →L[ℝ] E →L[ℝ] ℝ) (v : E) {ι : Type*}
+    (t : Finset ι) (d : ι → ℝ) (w : ι → E) :
+    L v (∑ j ∈ t, d j • w j) = ∑ j ∈ t, d j • L v (w j) := by
+  simp_rw [map_sum, ContinuousLinearMap.map_smul]
+
+lemma sum_mul_right (L : E →L[ℝ] E →L[ℝ] ℝ) (v : E) {ι : Type*}
+    (t : Finset ι) (d : ι → ℝ) (w : ι → E) :
+    L v (∑ j ∈ t, d j • w j) = ∑ j ∈ t, d j * L v (w j) := by
+  rw [sum_smul_right L v t d w]
+  simp_rw [smul_eq_mul]
+
+lemma sum_smul_left_right (L : E →L[ℝ] E →L[ℝ] ℝ) {ι₁ ι₂ : Type*}
+    (s : Finset ι₁) (t : Finset ι₂)
+    (c : ι₁ → ℝ) (v : ι₁ → E) (d : ι₂ → ℝ) (w : ι₂ → E) :
+    L (∑ i ∈ s, c i • v i) (∑ j ∈ t, d j • w j) =
+      ∑ i ∈ s, ∑ j ∈ t, c i • d j • L (v i) (w j) := by
+  simp_rw [sum_smul_left, sum_smul_right, Finset.smul_sum]
+
+lemma sum_mul_left_right (L : E →L[ℝ] E →L[ℝ] ℝ) {ι₁ ι₂ : Type*}
+    (s : Finset ι₁) (t : Finset ι₂)
+    (c : ι₁ → ℝ) (v : ι₁ → E) (d : ι₂ → ℝ) (w : ι₂ → E) :
+    L (∑ i ∈ s, c i • v i) (∑ j ∈ t, d j • w j) =
+      ∑ i ∈ s, ∑ j ∈ t, c i * d j * L (v i) (w j) := by
+  rw [sum_smul_left_right L s t c v d w]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  simp_rw [smul_eq_mul, mul_assoc]
+
+lemma on_basis_expansions
+    (b : Basis (Fin (Module.finrank ℝ E)) ℝ E)
+    (L : E →L[ℝ] E →L[ℝ] ℝ) (v w : E) :
+    L v w = ∑ i ∈ Finset.univ, ∑ j ∈ Finset.univ,
+      (b.coord i v) * (b.coord j w) * L (b i) (b j) := by
+  rw [vector_basis_expansion b v, vector_basis_expansion b w]
+  rw [sum_mul_left_right L Finset.univ Finset.univ
+    (fun i => b.coord i v) b (fun j => b.coord j w) b]
+  simp only [←vector_basis_expansion b v, ←vector_basis_expansion b w]
+
+/-- The sum of elementary bilinear forms weighted by coefficients,
+    when applied to vectors, equals a weighted sum of the products of
+    coordinate functions. -/
+lemma elementary_bilinear_forms_sum_apply
+    (b : Basis (Fin (Module.finrank ℝ E)) ℝ E)
+    (e : (Fin (Module.finrank ℝ E)) → (Fin (Module.finrank ℝ E)) → (E →L[ℝ] E →L[ℝ] ℝ))
+    (h_e : ∀ i j v w, e i j v w = (b.coord i) v * (b.coord j) w)
+    (c : (Fin (Module.finrank ℝ E)) → (Fin (Module.finrank ℝ E)) → ℝ)
+    (v w : E) :
+    ((∑ i ∈ Finset.univ, ∑ j ∈ Finset.univ, c i j • e i j) v) w =
+    ∑ i ∈ Finset.univ, ∑ j ∈ Finset.univ, c i j * ((b.coord i) v * (b.coord j) w) := by
+  simp only [ContinuousLinearMap.sum_apply, ContinuousLinearMap.smul_apply]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  congr
+  exact h_e i j v w
+
+/-- A simple algebraic identity for rearranging terms in double sums with multiple scalars. -/
+lemma Finset.rearrange_double_sum_coefficients {α β R : Type*} [CommSemiring R]
+    {s : Finset α} {t : Finset β} {f : α → β → R} {g : α → β → R} {h : α → β → R} :
+    (∑ i ∈ s, ∑ j ∈ t, f i j * g i j * h i j) =
+    (∑ i ∈ s, ∑ j ∈ t, h i j * (f i j * g i j)) := by
+  apply Finset.sum_congr rfl; intro i _
+  apply Finset.sum_congr rfl; intro j _
+  ring
+
+/-- Any bilinear form can be decomposed as a sum of elementary bilinear forms
+    weighted by the form's values on basis elements. -/
+theorem decomposition [FiniteDimensional ℝ E] (b : Basis (Fin (Module.finrank ℝ E)) ℝ E) :
+    ∀ (L : E →L[ℝ] E →L[ℝ] ℝ),
+    ∃ (e : (Fin (Module.finrank ℝ E)) → (Fin (Module.finrank ℝ E)) → (E →L[ℝ] E →L[ℝ] ℝ)),
+    (∀ i j v w, e i j v w = (b.coord i) v * (b.coord j) w) ∧
+    L = ∑ i ∈ Finset.univ, ∑ j ∈ Finset.univ, L (b i) (b j) • e i j := by
+  intro L
+  obtain ⟨e, h_e⟩ := elementary_bilinear_forms_def b
+  have h_decomp : L = ∑ i ∈ Finset.univ, ∑ j ∈ Finset.univ, L (b i) (b j) • e i j := by
+    ext v w
+    have expansion := on_basis_expansions b L v w
+    have right_side := elementary_bilinear_forms_sum_apply b e h_e (λ i j => L (b i) (b j)) v w
+    rw [expansion, Finset.rearrange_double_sum_coefficients]
+    exact right_side.symm
+  exact ⟨e, ⟨h_e, h_decomp⟩⟩
+
+/-- Given two vectors `v` and `w`, `eval_vectors_continuousLinear v w` is the continuous linear map
+that evaluates a bilinear form `B` at `(v, w)`. -/
+def eval_vectors_continuousLinear (v w : E) :
+    (E →L[ℝ] E →L[ℝ] ℝ) →L[ℝ] ℝ :=
+  { toFun := fun B => B v w,
+    map_add' := fun f g => by simp [ContinuousLinearMap.add_apply],
+    map_smul' := fun c f => by simp [ContinuousLinearMap.smul_apply] }
+
+namespace ContinuousLinearMap
+
+lemma map_sum_clm {R S M M₂ : Type*}
+    [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂]
+    {σ : R →+* S} [Module R M] [Module S M₂] (f : M →ₛₗ[σ] M₂) {ι : Type*} (t : Finset ι)
+    (g : ι → M) : f (∑ i ∈ t, g i) = ∑ i ∈ t, f (g i) :=
+  _root_.map_sum f g t
+
+lemma map_finset_sum {𝕜 : Type*} [NontriviallyNormedField 𝕜]
+    {E F : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+    [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+    (f : E →L[𝕜] F) {ι : Type*} (s : Finset ι) (g : ι → E) :
+    f (∑ i ∈ s, g i) = ∑ i ∈ s, f (g i) :=
+  _root_.map_sum f g s
+
+end ContinuousLinearMap

From a7059af449639460fe16dea16fb496aabbbad69e Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Fri, 9 May 2025 19:29:11 +0200
Subject: [PATCH 27/32] fix lints

---
 .../Geometry/Manifold/Chart/BilinearSmoothness.lean           | 4 ++--
 PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean          | 2 +-
 2 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Manifold/Chart/BilinearSmoothness.lean b/PhysLean/Mathematics/Geometry/Manifold/Chart/BilinearSmoothness.lean
index 8d928ce16..a5fd7fda9 100644
--- a/PhysLean/Mathematics/Geometry/Manifold/Chart/BilinearSmoothness.lean
+++ b/PhysLean/Mathematics/Geometry/Manifold/Chart/BilinearSmoothness.lean
@@ -47,8 +47,8 @@ lemma contDiffWithinAt_eval_bilinear_apply (hf : ContDiffWithinAt ℝ n f_bilin
 variable[FiniteDimensional ℝ E]
 
 lemma contDiffWithinAt_bilinear_apply_iff_forall_coord :
-  (∀ v w, ContDiffWithinAt ℝ n (fun x => f_bilin x v w) s_set x₀_pt) →
-  ContDiffWithinAt ℝ n f_bilin s_set x₀_pt := by
+    (∀ v w, ContDiffWithinAt ℝ n (fun x => f_bilin x v w) s_set x₀_pt) →
+    ContDiffWithinAt ℝ n f_bilin s_set x₀_pt := by
   intro h_comp
   rw [ContDiff.iff_forall_coord (V := E →L[ℝ] E →L[ℝ] ℝ) (𝕜 := ℝ)]
   intro φ
diff --git a/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean b/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean
index 1ed42aeaa..60442e788 100644
--- a/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean
+++ b/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean
@@ -111,7 +111,7 @@ lemma on_basis_expansions
       (b.coord i v) * (b.coord j w) * L (b i) (b j) := by
   rw [vector_basis_expansion b v, vector_basis_expansion b w]
   rw [sum_mul_left_right L Finset.univ Finset.univ
-    (fun i => b.coord i v) b (fun j => b.coord j w) b]
+      (fun i => b.coord i v) b (fun j => b.coord j w) b]
   simp only [←vector_basis_expansion b v, ←vector_basis_expansion b w]
 
 /-- The sum of elementary bilinear forms weighted by coefficients,

From b7d97b9fa5e472a986a31e7a5376f6842ec1c01d Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Mon, 12 May 2025 14:07:23 +0200
Subject: [PATCH 28/32] Update Defs.lean

---
 PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean | 8 +++++---
 1 file changed, 5 insertions(+), 3 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index 18e0abeec..ec4fbf467 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -5,7 +5,9 @@ Authors: Matteo Cipollina
 -/
 
 import Mathlib.Algebra.Lie.OfAssociative
-import Mathlib.MeasureTheory.Integral.IntervalIntegral
+import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
+import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
+import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts
 import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
 
 /-!
@@ -105,10 +107,10 @@ Riemannian metric. -/
 noncomputable def tangentInnerCore (g : RiemannianMetric I n M) (x : M) :
     InnerProductSpace.Core ℝ (TangentSpace I x) where
   inner := λ v w => g.inner x v w
-  conj_symm := λ v w => by
+  conj_inner_symm := λ v w => by
     simp only [inner_apply, conj_trivial]
     exact g.toPseudoRiemannianMetric.symm x w v
-  nonneg_re := λ v => by
+  re_inner_nonneg := λ v => by
     simp only [inner_apply, RCLike.re_to_real]
     by_cases hv : v = 0
     · simp [hv, inner_apply, map_zero]

From 13dea2ecee932552145f04caeac8525fc261341f Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Mon, 12 May 2025 14:16:36 +0200
Subject: [PATCH 29/32] Update BilinearForm.lean

---
 PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean b/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean
index 60442e788..029a7db87 100644
--- a/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean
+++ b/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean
@@ -112,7 +112,7 @@ lemma on_basis_expansions
   rw [vector_basis_expansion b v, vector_basis_expansion b w]
   rw [sum_mul_left_right L Finset.univ Finset.univ
       (fun i => b.coord i v) b (fun j => b.coord j w) b]
-  simp only [←vector_basis_expansion b v, ←vector_basis_expansion b w]
+  simp only [← vector_basis_expansion b v, ←vector_basis_expansion b w]
 
 /-- The sum of elementary bilinear forms weighted by coefficients,
     when applied to vectors, equals a weighted sum of the products of

From 63a4eba75f32ff6a5b0c33d3463458f9e9f845fb Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Mon, 12 May 2025 14:23:03 +0200
Subject: [PATCH 30/32] Update BilinearForm.lean

---
 PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean b/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean
index 029a7db87..06e7bc63b 100644
--- a/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean
+++ b/PhysLean/Mathematics/LinearAlgebra/BilinearForm.lean
@@ -112,7 +112,7 @@ lemma on_basis_expansions
   rw [vector_basis_expansion b v, vector_basis_expansion b w]
   rw [sum_mul_left_right L Finset.univ Finset.univ
       (fun i => b.coord i v) b (fun j => b.coord j w) b]
-  simp only [← vector_basis_expansion b v, ←vector_basis_expansion b w]
+  simp only [← vector_basis_expansion b v, ← vector_basis_expansion b w]
 
 /-- The sum of elementary bilinear forms weighted by coefficients,
     when applied to vectors, equals a weighted sum of the products of

From fe9d0fe648183aae86619529d6ed23a75a03859e Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Mon, 12 May 2025 14:57:25 +0200
Subject: [PATCH 31/32] Update Chart.lean

---
 .../Metric/PseudoRiemannian/Chart.lean        | 55 ++++++++++++-------
 1 file changed, 35 insertions(+), 20 deletions(-)

diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean
index 66c1c43bb..32e6ac0a1 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean
@@ -12,8 +12,12 @@ import PhysLean.Mathematics.LinearAlgebra.BilinearForm
 /-!
 # Pseudo-Riemannian Metric in Chart Coordinates
 
-This file defines `chartMetric`, which expresses a pseudo-Riemannian metric in local chart
-coordinates, and proves its transformation properties under coordinate changes.
+This file defines `chartMetric`, which expresses a pseudo-Riemannian metric `g`
+on a manifold `M` in local chart coordinates `e`. It establishes the
+tensor transformation law for `chartMetric` under a change of chart,
+`chartMetric_coord_change`. Auxiliary lemmas concern the smoothness of
+bilinear forms when viewed as maps into function spaces, and properties
+of chart transitions necessary for proving the main transformation result.
 -/
 
 namespace PseudoRiemannianMetric
@@ -24,10 +28,10 @@ open BilinearForm PartialHomeomorph ContinuousLinearMap PartialEquiv ContDiff Ma
 open Filter
 
 variable {E : Type v} {M : Type v} {n : WithTop ℕ∞}
-variable [NormedAddCommGroup E] [NormedSpace ℝ E] --[FiniteDimensional ℝ E]
-variable [TopologicalSpace M] [ChartedSpace E M] --[T2Space M]
-variable {I : ModelWithCorners ℝ E E} --[ModelWithCorners.Boundaryless I]
-variable [inst_mani_smooth : IsManifold I (n + 1) M] -- For C^{n+1} manifold for C^n metric
+variable [NormedAddCommGroup E] [NormedSpace ℝ E]
+variable [TopologicalSpace M] [ChartedSpace E M]
+variable {I : ModelWithCorners ℝ E E}
+variable [inst_mani_smooth : IsManifold I (n + 1) M]
 variable [inst_tangent_findim : ∀ (x : M), FiniteDimensional ℝ (TangentSpace I x)]
 
 noncomputable instance (x : M) : NormedAddCommGroup (TangentSpace I x) :=
@@ -36,9 +40,16 @@ noncomputable instance (x : M) : NormedAddCommGroup (TangentSpace I x) :=
 noncomputable instance (x : M) : NormedSpace ℝ (TangentSpace I x) :=
   show NormedSpace ℝ E from inferInstance
 
-/-- Given a pseudo-Riemannian metric `g` and a partial homeomorphism `e`, computes the metric
-in the chart coordinates. At a point `y` in the target of `e`, the result is the pullback of
-the metric at `e.symm y` by the derivative of `e.symm` at `y`. -/
+/-- Given a pseudo-Riemannian metric `g` on a manifold `M` and a chart `e : PartialHomeomorph M E`,
+`chartMetric g e y` computes the metric tensor in the local coordinates provided by `e`.
+Let `x := e.symm y` be the point on `M` corresponding to chart coordinates `y : E`.
+Let `Df_y := mfderiv I I e.symm y : E →L[ℝ] TangentSpace I x` be the differential of `e.symm` at
+`y`. Then `chartMetric g e y` is the bilinear form on `E` defined as the pullback of `g.val x` by
+`Df_y`. That is, for `v, w : E`, `(chartMetric g e y) v w = g.val x (Df_y v) (Df_y w)`.
+If `(b_i)` is a basis for `E`, then the values `(chartMetric g e y) (b_i) (b_j)` are the
+components `g_{ij}(y)` of the metric `g` at `x` with respect to the basis for `T_xM`
+induced by `(b_i)` via `Df_y`.
+If `y ∉ e.target`, the result is defined as the zero bilinear form. -/
 def chartMetric (g : PseudoRiemannianMetric E E M n I) (e : PartialHomeomorph M E) (y : E) :
     E →L[ℝ] E →L[ℝ] ℝ :=
   letI := Classical.propDecidable
@@ -50,6 +61,8 @@ def chartMetric (g : PseudoRiemannianMetric E E M n I) (e : PartialHomeomorph M
       BilinearForm.pullback (g.val x) Df)
     (fun _ => (0 : E →L[ℝ] E →L[ℝ] ℝ))
 
+/-- Evaluation of `chartMetric` at `y ∈ e.target`.
+For `v, w : E`, `chartMetric g e y v w = g_x(D(e⁻¹)_y v, D(e⁻¹)_y w)`, where `x = e⁻¹y`. -/
 @[simp]
 lemma chartMetric_apply_of_mem (g : PseudoRiemannianMetric E E M n I) (e : PartialHomeomorph M E)
     {y : E} (hy : y ∈ e.target) (v w : E) :
@@ -59,6 +72,7 @@ lemma chartMetric_apply_of_mem (g : PseudoRiemannianMetric E E M n I) (e : Parti
   simp only [BilinearForm.pullback, ContinuousLinearMap.bilinearComp_apply]
   exact rfl
 
+/-- If `y ∉ e.target`, `chartMetric g e y` is the zero bilinear form. -/
 lemma chartMetric_apply_of_not_mem
     (g : PseudoRiemannianMetric E E M n I)
     (e : PartialHomeomorph M E)
@@ -66,7 +80,10 @@ lemma chartMetric_apply_of_not_mem
     chartMetric g e y v w = 0 := by
   simp only [chartMetric, hy, ↓reduceDIte, ContinuousLinearMap.zero_apply]
 
-/-- The value of a metric in chart coordinates at corresponding points. -/
+/-- The metric `g_x(D(e⁻¹)_y v, D(e⁻¹)_y w)` can be expressed using a second chart `e'`
+via the chain rule for `D(e⁻¹) = D(e'⁻¹) ∘ D(φ)`.
+`x` is a point on the manifold, `y = ex`, `y' = e'x`.
+`φ = e' ∘ e⁻¹` is the transition map. `1 ≤ n`. -/
 lemma chartMetric_at_corresponding_points (g : PseudoRiemannianMetric E E M n I)
     (e e' : PartialHomeomorph M E) (x : M)
     (hx_e : x ∈ e.source) (hx_e' : x ∈ e'.source)
@@ -86,7 +103,9 @@ lemma chartMetric_at_corresponding_points (g : PseudoRiemannianMetric E E M n I)
   rw [h_chain]
   congr
 
-/-- The relationship between chart metrics under coordinate change. -/
+/-- Transformation law for the metric components under chart change.
+    `g_e(y)(v,w) = g_{e'}(y')(D(φ)_y v, D(φ)_y w)`, where `y = ex`, `y' = e'x`, `φ = e' ∘ e⁻¹`.
+    Assumes `e, e'` are in a `C^n` maximal atlas (`1 ≤ n`). -/
 lemma chartMetric_bilinear_pullback (g : PseudoRiemannianMetric E E M n I)
     (e e' : PartialHomeomorph M E) (x : M)
     (hx_e : x ∈ e.source) (hx_e' : x ∈ e'.source)
@@ -114,15 +133,11 @@ lemma chartMetric_bilinear_pullback (g : PseudoRiemannianMetric E E M n I)
   rw [h_phi_def]
   exact chartMetric_at_corresponding_points g e e' x hx_e hx_e' hmani hn1 he he' v w
 
-/--
-The metric tensor transformation law under chart changes.
-
-This theorem states that the representation of a pseudo-Riemannian metric in different charts
-are related by the pullback of the coordinate transformation between these charts.
-Specifically, if `e` and `e'` are two charts containing a point `x_pt`, then the metric
-expressed in chart `e` equals the pullback of the metric expressed in chart `e'`
-by the differential of the transition map from `e` to `e'`.
--/
+/-- Transformation law for `chartMetric`: `(g_e)_y = ((e'∘e⁻¹)^* (g_{e'})_{e'y})_y`.
+Given a metric `g`, and two charts `e, e'`, the representation of `g` in chart `e` at `y = ex`
+is the pullback by the transition map `φ = e' ∘ e⁻¹` (evaluated at `y`)
+of the representation of `g` in chart `e'` (evaluated at `y' = e'x = φy`).
+Assumes charts are `C^n` compatible, `1 ≤ n`. -/
 theorem chartMetric_coord_change (g : PseudoRiemannianMetric E E M n I)
     (e e' : PartialHomeomorph M E)
     (x_pt : M)

From 486cedf0f93c47e3f0a323b82c0f5ce465f2ab8b Mon Sep 17 00:00:00 2001
From: Matteo Cipollina <matteo.cipollina@yahoo.it>
Date: Mon, 12 May 2025 15:59:24 +0200
Subject: [PATCH 32/32] fix lint

---
 PhysLean.lean                                                   | 2 +-
 .../Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean     | 2 +-
 PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean       | 2 +-
 3 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/PhysLean.lean b/PhysLean.lean
index 5346f18f0..621f78fac 100644
--- a/PhysLean.lean
+++ b/PhysLean.lean
@@ -18,11 +18,11 @@ import PhysLean.Electromagnetism.Homogeneous
 import PhysLean.Electromagnetism.LorentzAction
 import PhysLean.Electromagnetism.MaxwellEquations
 import PhysLean.Electromagnetism.Wave
+import PhysLean.Mathematics.LinearAlgebra.BilinearForm
 import PhysLean.Mathematics.Analysis.ContDiff
 import PhysLean.Mathematics.FDerivCurry
 import PhysLean.Mathematics.Fin
 import PhysLean.Mathematics.Fin.Involutions
-import PhysLean.Mathematics.LinearAlgebra.BilinearForm
 import PhysLean.Mathematics.Geometry.Manifold.PartialHomeomorph
 import PhysLean.Mathematics.Geometry.Manifold.Chart.Utilities
 import PhysLean.Mathematics.Geometry.Manifold.Chart.BilinearSmoothness
diff --git a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean
index 32e6ac0a1..3bd811af9 100644
--- a/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/PseudoRiemannian/Chart.lean
@@ -5,9 +5,9 @@ Authors: Matteo Cipollina
 -/
 
 import Mathlib.Algebra.Lie.OfAssociative
+import PhysLean.Mathematics.LinearAlgebra.BilinearForm
 import PhysLean.Mathematics.Geometry.Manifold.Chart.Smoothness
 import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
-import PhysLean.Mathematics.LinearAlgebra.BilinearForm
 
 /-!
 # Pseudo-Riemannian Metric in Chart Coordinates
diff --git a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index ec4fbf467..2974fb6af 100644
--- a/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/PhysLean/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -8,7 +8,7 @@ import Mathlib.Algebra.Lie.OfAssociative
 import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
 import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
 import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts
-import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Defs
+import PhysLean.Mathematics.Geometry.Metric.PseudoRiemannian.Chart
 
 /-!
 # Riemannian Metric Definitions