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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 | import Mathlib
import OAI.Analysis.BiholderTransport.Geodesics.SprayDistanceBound
namespace OAI
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff
namespace WeakMTWTransport
variable {n : ℕ} {M : Type*} [MetricSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
lemma tangent_chart_trivialization {a x : M}
(hx : x∈(extChartAt 𝓘(ℝ,Model n) a).source) (p : TangentSpace 𝓘(ℝ,Model n) x) :
extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n))
(⟨a,0⟩ : TangentBundle 𝓘(ℝ,Model n) M) ⟨x,p⟩=
(extChartAt 𝓘(ℝ,Model n) a x,
(trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).continuousLinearMapAt ℝ x p) := by
rw [tangent_chart_apply,TangentBundle.continuousLinearMapAt_trivializationAt_eq_core
(by simpa only [extChartAt_source] using hx)]
rfl
lemma tangent_chart_symm_trivialization {a : M} {q : Model n×Model n}
(hq : q.1∈(extChartAt 𝓘(ℝ,Model n) a).target) :
(extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n))
(⟨a,0⟩ : TangentBundle 𝓘(ℝ,Model n) M)).symm q=
(⟨(extChartAt 𝓘(ℝ,Model n) a).symm q.1,
(trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ
((extChartAt 𝓘(ℝ,Model n) a).symm q.1) q.2⟩ : TangentBundle 𝓘(ℝ,Model n) M) := by
have hx : (extChartAt 𝓘(ℝ,Model n) a).symm q.1∈(chartAt (Model n) a).source := by
simpa only [extChartAt_source] using (extChartAt 𝓘(ℝ,Model n) a).map_target hq
rw [TangentBundle.symmL_trivializationAt_eq_core hx]
rfl
variable [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
def minimizingChartParameters (a : M) (K : Set M) : Set (Model n×Model n) :=
(extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) (⟨a,0⟩ : TangentBundle 𝓘(ℝ,Model n) M)) ''
{z : TangentBundle 𝓘(ℝ,Model n) M | z.2∈minimizingVectors z.1 ∧ z.1∈K}
section
variable [CompactSpace M]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
lemma isCompact_minimizingChartParameters {a : M} {K : Set M}
(hK : IsCompact K) (hKs : K⊆(extChartAt 𝓘(ℝ,Model n) a).source) :
IsCompact (minimizingChartParameters (n := n) a K) := by
have hc : IsCompact {z : TangentBundle 𝓘(ℝ,Model n) M | z.2∈minimizingVectors z.1 ∧ z.1∈K} :=
(isCompact_total_minimizingVectors (n := n) (M := M)).inter_right
(hK.isClosed.preimage (FiberBundle.continuous_proj _ _))
exact hc.image_of_continuousOn ((continuousOn_extChartAt _).mono (fun z hz =>
(tangent_chart_source_iff _ z).mpr (hKs hz.2)))
end
lemma mem_minimizingChartParameters {a : M} {K : Set M}
(hKs : K⊆(extChartAt 𝓘(ℝ,Model n) a).source) {q : Model n×Model n}
(hq : q∈minimizingChartParameters (n := n) a K) :
q.1∈(extChartAt 𝓘(ℝ,Model n) a).target ∧
((extChartAt 𝓘(ℝ,Model n) a).symm q.1∈K ∧
(trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ
((extChartAt 𝓘(ℝ,Model n) a).symm q.1) q.2∈
minimizingVectors ((extChartAt 𝓘(ℝ,Model n) a).symm q.1)) := by
obtain ⟨z,hz,rfl⟩ := hq
let χ := extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n))
(⟨a,0⟩ : TangentBundle 𝓘(ℝ,Model n) M)
have hzs : z∈χ.source := (tangent_chart_source_iff _ z).mpr (hKs hz.2)
have hb : (χ z).1∈(extChartAt 𝓘(ℝ,Model n) a).target :=
(tangent_chart_target_iff _ _).mp (χ.map_source hzs)
refine ⟨hb,?_⟩
have H : (χ.symm (χ z)).1∈K ∧ (χ.symm (χ z)).2∈minimizingVectors (χ.symm (χ z)).1 := by
rw [χ.left_inv hzs]
exact ⟨hz.2,hz.1⟩
rw [tangent_chart_symm_trivialization hb] at H
exact H
lemma trivialization_mem_minimizingChartParameters {a x : M} {K : Set M}
(hKs : K⊆(extChartAt 𝓘(ℝ,Model n) a).source) (hx : x∈K)
{p : TangentSpace 𝓘(ℝ,Model n) x} (hp : p∈minimizingVectors x) :
(extChartAt 𝓘(ℝ,Model n) a x,
(trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).continuousLinearMapAt ℝ x p)∈
minimizingChartParameters (n := n) a K :=
⟨⟨x,p⟩,⟨hp,hx⟩,tangent_chart_trivialization (hKs hx) p⟩
end WeakMTWTransport
end
end OAI
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