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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 | import Mathlib
import OAI.Analysis.BiholderTransport.Geodesics.ExpInteriorUnique
import OAI.Analysis.BiholderTransport.Coordinates.ExpNonconjugacy
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
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Bundle
open scoped Topology Bundle
open scoped Topology
open Function Manifold Set
open Manifold Bundle
open scoped Manifold Bundle
open Set
open Set Filter
open scoped Topology ContDiff
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
namespace WeakMTWTransport
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
omit [CompactSpace M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M] in
lemma interior_minimizingVectors_subset (x : M) :
interior (minimizingVectors (n := n) x) ⊆ injectivityDomain x := by
intro p hp
have h : {a : ℝ | a • p ∈ minimizingVectors (n := n) x} ∈ 𝓝 (1:ℝ) := by
have H := (continuous_id.smul continuous_const : Continuous (fun a : ℝ => a • p))
apply H.continuousAt.preimage_mem_nhds
change minimizingVectors (n := n) x ∈ 𝓝 ((1:ℝ) • p)
simpa only [one_smul] using mem_interior_iff_mem_nhds.mp hp
obtain ⟨δ,hδ,hball⟩ := Metric.mem_nhds_iff.mp h
let a := 1+δ/2
have ha : 1<a := by dsimp [a]; linarith
have hpa : a • p ∈ minimizingVectors (n := n) x := by
apply hball
rw [Metric.mem_ball,Real.dist_eq]
dsimp [a]
rw [add_sub_cancel_left,abs_of_pos (by positivity : 0<δ/2)]
linarith
refine ⟨a,ha,?_⟩
change dist x (riemannianExp x (a • p)) = a*‖p‖
change dist x (riemannianExp x (a • p)) = ‖a • p‖ at hpa
simpa only [norm_smul,Real.norm_eq_abs,abs_of_pos (lt_trans zero_lt_one ha)] using hpa
lemma injectivityDomain_subset_interior_minimizingVectors (x : M) :
injectivityDomain (n := n) x ⊆ interior (minimizingVectors x) := by
intro p hp
obtain ⟨U,hU,hpU,hinj⟩ := riemannianExp_locally_injective hp
let K := minimizingVectors (n := n) x \ U
have hK : IsCompact K := (isCompact_minimizingVectors x).diff hU
have hclosed : IsClosed (riemannianExp (n := n) x '' K) :=
(hK.image (continuous_riemannianExp x)).isClosed
have hnot : riemannianExp x p ∉ riemannianExp x '' K := by
rintro ⟨q,hq,he⟩
have heq := riemannianExp_interior_unique hp hq.1 he.symm
exact hq.2 (heq ▸ hpU)
have hnear : ∀ᶠ v in 𝓝 p, v ∈ U ∧ riemannianExp x v ∉ riemannianExp x '' K :=
Filter.Eventually.and (hU.mem_nhds hpU)
((continuous_riemannianExp x).continuousAt.preimage_mem_nhds
(hclosed.isOpen_compl.mem_nhds hnot))
apply mem_interior_iff_mem_nhds.mpr
filter_upwards [hnear] with v hv
obtain ⟨q,hq,he⟩ := exists_minimizing_vector (n := n) x (riemannianExp x v)
have hqU : q ∈ U := by
by_contra hqU
exact hv.2 ⟨q,⟨hq,hqU⟩,he⟩
have hvq := hinj hv.1 hqU he.symm
exact hvq ▸ hq
lemma injectivityDomain_eq_interior_minimizingVectors (x : M) :
injectivityDomain (n := n) x = interior (minimizingVectors x) :=
Set.Subset.antisymm (injectivityDomain_subset_interior_minimizingVectors x)
(interior_minimizingVectors_subset x)
lemma isOpen_injectivityDomain (x : M) : IsOpen (injectivityDomain (n := n) x) := by
rw [injectivityDomain_eq_interior_minimizingVectors]
exact isOpen_interior
end WeakMTWTransport
end
end OAI
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