| import Mathlib |
| import OAI.Analysis.BiholderTransport.Coordinates.Coordinates |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology |
| open MeasureTheory |
| open scoped ContDiff ENNReal |
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| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory |
| open scoped Topology ContDiff Manifold Bundle ENNReal |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology Manifold |
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| namespace WeakMTWTransport |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| local instance smoothDistanceNormedSpace : NormedSpace ℝ E := |
| InnerProductSpace.toNormedSpace |
| local instance smoothDistanceNormedAddCommGroupEndomorphism : |
| NormedAddCommGroup (E →L[ℝ] E) := ContinuousLinearMap.toNormedAddCommGroup |
| local instance smoothDistanceNormedSpaceEndomorphism : |
| NormedSpace ℝ (E →L[ℝ] E) := ContinuousLinearMap.toNormedSpace |
| local instance smoothDistanceNormedAddCommGroupDual : |
| NormedAddCommGroup (E →L[ℝ] ℝ) := ContinuousLinearMap.toNormedAddCommGroup |
| local instance smoothDistanceNormedSpaceDual : |
| NormedSpace ℝ (E →L[ℝ] ℝ) := ContinuousLinearMap.toNormedSpace |
| local instance smoothDistanceNormedAddCommGroupBilinear : |
| NormedAddCommGroup (E →L[ℝ] E →L[ℝ] ℝ) := ContinuousLinearMap.toNormedAddCommGroup |
| local instance smoothDistanceNormedSpaceBilinear : |
| NormedSpace ℝ (E →L[ℝ] E →L[ℝ] ℝ) := ContinuousLinearMap.toNormedSpace |
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| lemma derivative_pullback_metric |
| {g h : E → E →L[ℝ] E →L[ℝ] ℝ} {φ : E → E} {x : E} |
| (hg : DifferentiableAt ℝ g x) (hh : DifferentiableAt ℝ h (φ x)) |
| (hφ : DifferentiableAt ℝ φ x) (hDφ : DifferentiableAt ℝ (fderiv ℝ φ) x) |
| (hmetric : ∀ᶠ y in 𝓝 x, ∀ u v, g y u v = h (φ y) (fderiv ℝ φ y u) (fderiv ℝ φ y v)) |
| (w u v : E) : |
| fderiv ℝ g x w u v = |
| fderiv ℝ h (φ x) (fderiv ℝ φ x w) (fderiv ℝ φ x u) (fderiv ℝ φ x v) + |
| h (φ x) (fderiv ℝ (fderiv ℝ φ) x w u) (fderiv ℝ φ x v) + |
| h (φ x) (fderiv ℝ φ x u) (fderiv ℝ (fderiv ℝ φ) x w v) := by |
| have hleft := (hg.hasFDerivAt.clm_apply (hasFDerivAt_const u x)).clm_apply |
| (hasFDerivAt_const v x) |
| have hright := ((hh.hasFDerivAt.comp x hφ.hasFDerivAt).clm_apply |
| (hDφ.hasFDerivAt.clm_apply (hasFDerivAt_const u x))).clm_apply |
| (hDφ.hasFDerivAt.clm_apply (hasFDerivAt_const v x)) |
| have heq : (fun y => g y u v) =ᶠ[𝓝 x] |
| (fun y => h (φ y) (fderiv ℝ φ y u) (fderiv ℝ φ y v)) := by |
| filter_upwards [hmetric] with y hy; exact hy u v |
| have he := congrArg (fun L : E →L[ℝ] ℝ => L w) |
| ((hleft.congr_of_eventuallyEq heq.symm).unique hright) |
| simpa [add_assoc,add_comm,add_left_comm] using he |
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| lemma christoffel_pushforward_pairing |
| {g h : E → E →L[ℝ] E →L[ℝ] ℝ} {φ : E → E} {x : E} |
| (hg : DifferentiableAt ℝ g x) (hh : DifferentiableAt ℝ h (φ x)) |
| (hφ : ContDiffAt ℝ ∞ φ x) |
| (hmetric : ∀ᶠ y in 𝓝 x, ∀ u v, g y u v = h (φ y) (fderiv ℝ φ y u) (fderiv ℝ φ y v)) |
| (hgi : (g x).IsInvertible) (hhi : (h (φ x)).IsInvertible) |
| (hhs : ∀ u v, h (φ x) u v = h (φ x) v u) (u v w : E) : |
| h (φ x) (fderiv ℝ φ x (coordinateChristoffel g x u v)) (fderiv ℝ φ x w) = |
| h (φ x) (fderiv ℝ (fderiv ℝ φ) x u v + |
| coordinateChristoffel h (φ x) (fderiv ℝ φ x u) (fderiv ℝ φ x v)) |
| (fderiv ℝ φ x w) := by |
| have hpd := hφ.differentiableAt (by simp) |
| have hD := (hφ.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp) |
| have hp := derivative_pullback_metric hg hh hpd hD hmetric |
| have hm := hmetric.self_of_nhds |
| have hb : ∀ u v, fderiv ℝ (fderiv ℝ φ) x u v = fderiv ℝ (fderiv ℝ φ) x v u := by |
| intro u v |
| exact (hφ.isSymmSndFDerivAt (by simp)).eq u v |
| have h1 := coordinateChristoffel_metric g x u v w hgi |
| have h2 := coordinateChristoffel_metric h (φ x) (fderiv ℝ φ x u) |
| (fderiv ℝ φ x v) (fderiv ℝ φ x w) hhi |
| rw [hm,hp u v w,hp v u w,hp w u v] at h1 |
| rw [hb v u,hb w u,hb w v] at h1 |
| rw [hhs (fderiv ℝ φ x v) _,hhs (fderiv ℝ φ x u) _] at h1 |
| simp only [map_add,add_apply] |
| linarith |
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| lemma christoffel_pushforward |
| {g h : E → E →L[ℝ] E →L[ℝ] ℝ} {φ : E → E} {x : E} |
| (hg : DifferentiableAt ℝ g x) (hh : DifferentiableAt ℝ h (φ x)) |
| (hφ : ContDiffAt ℝ ∞ φ x) (hφs : Function.Surjective (fderiv ℝ φ x)) |
| (hmetric : ∀ᶠ y in 𝓝 x, ∀ u v, g y u v = h (φ y) (fderiv ℝ φ y u) (fderiv ℝ φ y v)) |
| (hgi : (g x).IsInvertible) (hhi : (h (φ x)).IsInvertible) |
| (hhs : ∀ u v, h (φ x) u v = h (φ x) v u) (u v : E) : |
| fderiv ℝ φ x (coordinateChristoffel g x u v) = |
| fderiv ℝ (fderiv ℝ φ) x u v + |
| coordinateChristoffel h (φ x) (fderiv ℝ φ x u) (fderiv ℝ φ x v) := by |
| have heq : h (φ x) (fderiv ℝ φ x (coordinateChristoffel g x u v)) = |
| h (φ x) (fderiv ℝ (fderiv ℝ φ) x u v + |
| coordinateChristoffel h (φ x) (fderiv ℝ φ x u) (fderiv ℝ φ x v)) := by |
| ext w |
| obtain ⟨z,rfl⟩ := hφs w |
| exact christoffel_pushforward_pairing hg hh hφ hmetric hgi hhi hhs u v z |
| rcases hhi with ⟨A,hA⟩ |
| apply A.injective |
| change A.toContinuousLinearMap _ = A.toContinuousLinearMap _ |
| rw [hA] |
| exact heq |
| end WeakMTWTransport |
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| end |
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| end OAI |
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