feat: metric connections

Mathlib.lean

@@ -4467,6 +4467,7 @@ public import Mathlib.Geometry.Manifold.SmoothEmbedding
 public import Mathlib.Geometry.Manifold.StructureGroupoid
 public import Mathlib.Geometry.Manifold.VectorBundle.Basic
 public import Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic
+public import Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Metric
 public import Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Torsion
 public import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
 public import Mathlib.Geometry.Manifold.VectorBundle.Hom

Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean

@@ -790,6 +790,20 @@ theorem mfderiv_add (hf : MDiffAt f z) (hg : MDiffAt g z) :
       (by exact mfderiv% f z) + (by exact mfderiv% g z) :=
   (hf.hasMFDerivAt.add hg.hasMFDerivAt).mfderiv
 
+open NormedSpace in
+theorem fromTangentSpace_mfderiv_add (hf : MDiffAt f z) (hg : MDiffAt g z) :
+    (fromTangentSpace _).toContinuousLinearMap ∘L (mfderiv% (f + g) z)
+    = (fromTangentSpace _).toContinuousLinearMap ∘L (mfderiv% f z)
+    + (fromTangentSpace _).toContinuousLinearMap ∘L (mfderiv% g z) :=
+  (hf.hasMFDerivAt.add hg.hasMFDerivAt).mfderiv
+
+open NormedSpace in
+theorem fromTangentSpace_mfderiv_add_apply (hf : MDiffAt f z) (hg : MDiffAt g z)
+    (v : TangentSpace I x) :
+    fromTangentSpace _ (mfderiv% (f + g) z v)
+    = fromTangentSpace _ (mfderiv% f z v) + fromTangentSpace _ (mfderiv% g z v) :=
+  congr($(fromTangentSpace_mfderiv_add hf hg) v)
+
 section sum
 variable {ι : Type} {t : Finset ι} {f : ι → M → E'} {f' : ι → TangentSpace I z →L[𝕜] E'}
 

Mathlib/Geometry/Manifold/VectorBundle/CovariantDerivative/Metric.lean

@@ -0,0 +1,281 @@
+/-
+Copyright (c) 2025 Michael Rothgang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Patrick Massot, Michael Rothgang, Heather Macbeth
+-/
+module
+
+public import Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic
+public import Mathlib.Geometry.Manifold.VectorBundle.Riemannian
+public import Mathlib.Geometry.Manifold.MFDeriv.NormedSpace
+
+/-! # Metric connections
+
+This file defines connections on a Riemannian vector bundle which are compatible with the ambient
+metric. A bundled connection `∇` on a Riemannian vector bundle `(V, g)` is compatible with the
+metric `g` if and only if the differentiated metric tensor `∇ g` (defined by
+`(X, σ, τ) ↦ ∇_X g(σ, τ) - g(∇_X σ, τ) - g(σ, ∇_X τ)`) vanishes on all differentiable vector fields
+`X` and differentiable sections `σ`, `τ`.
+
+## Main definitions and results
+
+* `CovariantDerivative.compatibilityTensor`: the tensor
+  `(X, σ, τ) ↦ X g(σ, τ) - g(∇_X σ, τ) - g(σ, ∇_X τ)` defining when a connection `∇` on a Riemannian
+  vector bundle `(V, g)` is compatible with the metric `g`.
+* `CovariantDerivative.compatibilityTensor_apply` and
+  `CovariantDerivative.compatibilityTensor_apply` give formulas for applying the compatibility
+  tensor at `x` to vector fields and sections which are differentiable at `x`,
+  resp. to extensions of tangent vectors and sections at `x` to differentiable vector fields and
+  sections near `x`.
+* `CovariantDerivative.IsCompatible`: predicate for a connection to be metric, namely that
+  `∇` is metric iff its `compatibilityTensor` vanishes
+
+## TODO
+
+* When Mathlib has a notion of parallel transport, prove the equivalence of
+ `CovariantDerivative.IsCompatible` with the characterisation that parallel transport be an
+  isometry.
+
+* Given connections on bundles `V` and `W`, there is an induced connnection on the bundle Hom(V, W).
+  When this induced connection has been defined in Mathlib, rephrase the definition of
+  `CovariantDerivative.compatibilityTensor`, to be simply the covariant derivative of the metric
+  tensor (considered as a section of Hom(V, Hom(V, ℝ))).
+
+-/
+open Bundle NormedSpace
+open scoped Manifold ContDiff
+
+@[expose] public section
+
+variable
+  -- Let `M` be a `C^k` real manifold modeled on `(E, H)`
+  {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  {H : Type*} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)
+  {M : Type*}
+  -- Let `V` be a bundle over `M`, endowed with a Riemannian metric.
+  (F : Type*) [NormedAddCommGroup F] [NormedSpace ℝ F]
+  {V : M → Type*} [TopologicalSpace (TotalSpace F V)]
+  [∀ x, NormedAddCommGroup (V x)] [∀ x, InnerProductSpace ℝ (V x)]
+
+/-! # Compatible connections
+
+A connection on `V` is compatible with the metric on `V` iff `X ⟨σ, τ⟩ = ⟨∇_X σ, τ⟩ + ⟨σ, ∇_X τ⟩`
+holds for all sufficiently nice vector fields `X` on `M` and sections `σ`, `τ` of `V`.
+The left hand side is the pushforward of the function `⟨σ, τ⟩` along the vector field `X`:
+the left hand side at `x` is `df(X x)`, where `f := ⟨σ, τ⟩` (ie. `X` is seen a derivation on
+the algebra of function on the base manifold acting on the function `⟨σ, τ⟩`).
+In our definition, we ask for this identity to at each `x : M`, whenever `X`, `σ` and `τ` are
+differentiable at `x`.
+-/
+
+variable {σ σ' σ'' τ τ' τ'' : Π x : M, V x}
+
+-- set_option trace.profiler true
+-- set_option profiler.threshold 500
+
+/-- The scalar product of two sections. -/
+noncomputable abbrev product (σ τ : Π x : M, V x) : M → ℝ :=
+  fun x ↦ inner ℝ (σ x) (τ x)
+
+-- `product` is C^k if σ and τ are: this is shown in `Riemannian.lean`
+
+local notation "⟪" σ ", " τ "⟫" => product σ τ
+
+-- Basic API for the product of two sections.
+section product
+
+lemma product_apply (x) : ⟪σ, τ⟫ x = inner ℝ (σ x) (τ x) := rfl
+
+variable (σ σ' τ)
+
+lemma product_swap : ⟪τ, σ⟫ = ⟪σ, τ⟫ := by
+  ext x
+  apply real_inner_comm
+
+@[simp]
+lemma product_zero_left : ⟪0, σ⟫ = 0 := by
+  ext x
+  simp only [product, Pi.zero_apply, inner_zero_left]
+
+@[simp]
+lemma product_zero_right : ⟪σ, 0⟫ = 0 := by rw [product_swap, product_zero_left]
+
+lemma product_add_left : ⟪σ + σ', τ⟫ = ⟪σ, τ⟫ + ⟪σ', τ⟫ := by
+  ext x
+  simp [product, InnerProductSpace.add_left]
+
+@[simp]
+lemma product_add_left_apply (x) : ⟪σ + σ', τ⟫ x = ⟪σ, τ⟫ x + ⟪σ', τ⟫ x := by
+  simp [product, InnerProductSpace.add_left]
+
+lemma product_add_right : ⟪σ, τ + τ'⟫ = ⟪σ, τ⟫ + ⟪σ, τ'⟫ := by
+  rw [product_swap, product_swap τ, product_swap τ', product_add_left]
+
+@[simp]
+lemma product_add_right_apply (x) : ⟪σ, τ + τ'⟫ x = ⟪σ, τ⟫ x + ⟪σ, τ'⟫ x := by
+  rw [product_swap, product_swap τ, product_swap τ', product_add_left_apply]
+
+@[simp] lemma product_neg_left : ⟪-σ, τ⟫ = -⟪σ, τ⟫ := by ext x; simp [product]
+
+@[simp] lemma product_neg_right : ⟪σ, -τ⟫ = -⟪σ, τ⟫ := by ext x; simp [product]
+
+lemma product_sub_left : ⟪σ - σ', τ⟫ = ⟪σ, τ⟫ - ⟪σ', τ⟫ := by
+  ext x
+  simp [product, inner_sub_left]
+
+lemma product_sub_right : ⟪σ, τ - τ'⟫ = ⟪σ, τ⟫ - ⟪σ, τ'⟫ := by
+  ext x
+  simp [product, inner_sub_right]
+
+lemma product_smul_left (f : M → ℝ) : product (f • σ) τ = f • product σ τ := by
+  ext x
+  simp [product, real_inner_smul_left]
+
+@[simp]
+lemma product_smul_const_left (a : ℝ) : product (a • σ) τ = a • product σ τ := by
+  ext x
+  simp [product, real_inner_smul_left]
+
+lemma product_smul_right (f : M → ℝ) : product σ (f • τ) = f • product σ τ := by
+  ext x
+  simp [product, real_inner_smul_right]
+
+@[simp]
+lemma product_smul_const_right (a : ℝ) : product σ (a • τ) = a • product σ τ := by
+  ext x
+  simp [product, real_inner_smul_right]
+
+end product
+
+-- These lemmas are necessary as my Lie bracket identities (assuming minimal differentiability)
+-- only hold point-wise. They abstract the expanding and unexpanding of `product`.
+lemma product_congr_left {x} (h : σ x = σ' x) : product σ τ x = product σ' τ x := by
+  rw [product_apply, h, ← product_apply]
+
+lemma product_congr_left₂ {x} (h : σ x = σ' x + σ'' x) :
+    product σ τ x = product σ' τ x + product σ'' τ x := by
+  rw [product_apply, h, inner_add_left, ← product_apply]
+
+lemma product_congr_right {x} (h : τ x = τ' x) : product σ τ x = product σ τ' x := by
+  rw [product_apply, h, ← product_apply]
+
+lemma product_congr_right₂ {x} (h : τ x = τ' x + τ'' x) :
+    product σ τ x = product σ τ' x + product σ τ'' x := by
+  rw [product_apply, h, inner_add_right, ← product_apply]
+
+namespace CovariantDerivative
+
+-- Let `cov` be a covariant derivative on `V`.
+-- TODO: include in cheat sheet!
+variable [TopologicalSpace M] [ChartedSpace H M] [FiberBundle F V]
+  (cov : CovariantDerivative I F V)
+
+/-- Local notation for a covariant derivative on a vector bundle acting on a vector field and a
+section. -/
+local syntax "∇" term:arg term : term
+local macro_rules | `(∇ $X $σ) => `(fun (x : M) ↦ cov $σ x ($X x))
+
+variable {F}
+
+/-- The function defining the compatibility tensor for `∇` w.r.t. `g`:
+prefer using `compatibilityTensor` instead -/
+noncomputable def compatibilityTensorAux (σ τ : Π x : M, V x) :
+    Π (x : M), TangentSpace I x →L[ℝ] ℝ := fun x ↦
+  (NormedSpace.fromTangentSpace _).toContinuousLinearMap ∘L mfderiv% ⟪σ, τ⟫ x
+  - ((innerSL ℝ (τ x)) ∘L (cov σ x)) - ((innerSL ℝ (σ x)) ∘L (cov τ x))
+
+lemma compatibilityTensorAux_apply (σ τ : Π x : M, V x)
+    {x : M} (X₀ : TangentSpace I x) :
+    compatibilityTensorAux I cov σ τ x X₀ =
+      NormedSpace.fromTangentSpace _ (mfderiv% ⟪σ, τ⟫ x X₀)
+      - inner ℝ (cov σ x X₀) (τ x) - inner ℝ (σ x) (cov τ x X₀) := by
+  rw [real_inner_comm]
+  rfl
+
+variable [VectorBundle ℝ F V] [IsContMDiffRiemannianBundle I 1 F V] {x : M}
+
+theorem compatibilityTensorAux_tensorial₁ (τ : Π x, V x) (hτ : MDiffAt (T% τ) x) :
+    TensorialAt I F (compatibilityTensorAux I cov · τ x) x where
+  smul hf hσ := by
+    ext X₀
+    rw [compatibilityTensorAux_apply, product_smul_left,
+      fromTangentSpace_mfderiv_smul_apply hf (hσ.inner_bundle hτ)]
+    simp [compatibilityTensorAux_apply, cov.isCovariantDerivativeOn.leibniz hσ hf, inner_add_left,
+      inner_smul_left]
+    ring
+  add hσ hσ' := by
+    ext X₀
+    rw [compatibilityTensorAux_apply, product_add_left,
+      fromTangentSpace_mfderiv_add_apply (hσ.inner_bundle hτ) (hσ'.inner_bundle hτ)]
+    simp [compatibilityTensorAux_apply, cov.isCovariantDerivativeOn.add hσ hσ', inner_add_left]
+    abel
+
+theorem compatibilityTensorAux_tensorial₂ (σ : Π x, V x) (hσ : MDiffAt (T% σ) x) :
+    TensorialAt I F (compatibilityTensorAux I cov σ · x) x where
+  smul hf hτ := by
+    ext X₀
+    rw [compatibilityTensorAux_apply, product_smul_right,
+      fromTangentSpace_mfderiv_smul_apply hf (hσ.inner_bundle hτ)]
+    simp [compatibilityTensorAux_apply, cov.isCovariantDerivativeOn.leibniz hτ hf, inner_add_right,
+      inner_smul_right]
+    ring
+  add hτ hτ' := by
+    ext X₀
+    rw [compatibilityTensorAux_apply, product_add_right,
+      fromTangentSpace_mfderiv_add_apply (hσ.inner_bundle hτ) (hσ.inner_bundle hτ')]
+    simp [compatibilityTensorAux_apply, cov.isCovariantDerivativeOn.add hτ hτ', inner_add_right]
+    abel
+
+variable {I} [ContMDiffVectorBundle 1 F V I] in
+/-- The tensor `(X, σ, τ) ↦ X g(σ, τ) - g(∇_X σ, τ) - g(σ, ∇_X τ)` defining when a connection
+`∇` on a Riemannian bundle `(M, V)` is compatible with the metric `g`. -/
+@[no_expose] noncomputable def compatibilityTensor [FiniteDimensional ℝ F] (x : M) :
+    V x →L[ℝ] V x →L[ℝ] (TangentSpace I x →L[ℝ] ℝ) :=
+  TensorialAt.mkHom₂ (compatibilityTensorAux I cov · · x) _
+    (compatibilityTensorAux_tensorial₁ I cov) (compatibilityTensorAux_tensorial₂ I cov)
+
+variable {X : Π x : M, TangentSpace I x}
+
+variable {I} [ContMDiffVectorBundle 1 F V I] in
+theorem compatibilityTensor_apply [FiniteDimensional ℝ F] (x : M)
+    (hσ : MDiffAt (T% σ) x) (hτ : MDiffAt (T% τ) x) :
+    cov.compatibilityTensor x (σ x) (τ x) (X x) =
+    fromTangentSpace _ (mfderiv% ⟪σ, τ⟫ x (X x)) - ⟪∇ X σ, τ⟫ x - ⟪σ, ∇ X τ⟫ x := by
+  unfold compatibilityTensor
+  rw [TensorialAt.mkHom₂_apply _ _ hσ hτ, compatibilityTensorAux_apply]
+
+open FiberBundle in
+variable {I} [ContMDiffVectorBundle 1 F V I] in
+theorem compatibilityTensor_apply_eq_extend [FiniteDimensional ℝ F] (X₀ : TangentSpace I x)
+    (σ₀ τ₀ : V x) :
+    cov.compatibilityTensor x σ₀ τ₀ X₀ =
+      fromTangentSpace _ (mfderiv% ⟪(extend F σ₀), (extend F τ₀)⟫ x X₀)
+        - inner ℝ (cov (extend F σ₀) x X₀) τ₀
+        - inner ℝ σ₀ (cov (extend F τ₀) x X₀) := by
+  simp [compatibilityTensor, TensorialAt.mkHom₂_apply_eq_extend, compatibilityTensorAux_apply]
+
+variable {I} [ContMDiffVectorBundle 1 F V I] in
+/-- Predicate saying that a connection `∇` on a Riemannian bundle `(V, g)` is compatible with the
+ambient metric, i.e. for all differentiable vector fields `X` on `M` and sections `σ` and `τ` of
+`V`, we have `X ⟨σ, τ⟩ = ⟨∇_X σ, τ⟩ + ⟨σ, ∇_X τ⟩`. -/
+def IsCompatible [FiniteDimensional ℝ F] : Prop := compatibilityTensor cov = 0
+
+variable {I} [IsManifold I 1 M] [ContMDiffVectorBundle 1 F V I]
+
+open FiberBundle in
+lemma isCompatible_iff [FiniteDimensional ℝ F] :
+    cov.IsCompatible ↔ ∀ {x : M} {X : Π x, TangentSpace I x} {σ τ : (x : M) → V x},
+      MDiffAt (T% X) x → MDiffAt (T% σ) x → MDiffAt (T% τ) x →
+      fromTangentSpace _ (mfderiv% ⟪σ, τ⟫ x (X x)) = ⟪∇ X σ, τ⟫ x + ⟪σ, ∇ X τ⟫ x := by
+  refine ⟨fun hcov x X σ τ hX hσ hτ ↦ ?_, fun h ↦ ?_⟩
+  · have H := congr($hcov x (σ x) (τ x) (X x))
+    simp [compatibilityTensor_apply _ _ hσ hτ] at H
+    linear_combination H
+  ext x σ₀ τ₀ X₀
+  rw [compatibilityTensor_apply_eq_extend]
+  have h' := h (mdifferentiableAt_extend I E X₀) (mdifferentiableAt_extend I F σ₀)
+    (mdifferentiableAt_extend I F τ₀)
+  simp [product] at h' ⊢
+  linear_combination h'
+
+end CovariantDerivative