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110f4f3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | import Mathlib
import OAI.Analysis.BiholderTransport.Coordinates.Coordinates
namespace OAI
noncomputable section
open Set MeasureTheory Manifold Bundle
open scoped ContDiff Manifold ENNReal NNReal Topology
open Set Filter
open scoped Topology NNReal
open Set Filter
open scoped Topology
open Set Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set
open scoped Topology
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter ContinuousLinearMap InnerProductSpace
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open scoped NNReal
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology
open MeasureTheory
open scoped ContDiff ENNReal
open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory
open scoped Topology ContDiff Manifold Bundle ENNReal
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology Manifold
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
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
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
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
end
end OAI
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