| import Mathlib |
| import OAI.Analysis.BiholderTransport.Geodesics.ExpInteriorUnique |
| import OAI.Analysis.BiholderTransport.Coordinates.ExpNonconjugacy |
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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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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Bundle |
| open scoped Topology Bundle |
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| open scoped Topology |
| open Function Manifold Set |
| open Manifold Bundle |
| open scoped Manifold Bundle |
| open Set |
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| open Set Filter |
| open scoped Topology ContDiff |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| 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] |
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| 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 |
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| 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 |
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| 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) |
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| lemma isOpen_injectivityDomain (x : M) : IsOpen (injectivityDomain (n := n) x) := by |
| rw [injectivityDomain_eq_interior_minimizingVectors] |
| exact isOpen_interior |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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