| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Geodesics.MinimizingChartParameters | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Manifold Bundle Module | |
| open scoped Topology ContDiff | |
| 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] | |
| local instance tangentFiniteNormalInjective (x : M) : | |
| FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| end WeakMTWTransport | |
| end | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| 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] | |
| def minimizingEndpointChartParameters (a c : M) (K L : Set M) : | |
| Set (Model n×(Model n×Model n)) := | |
| (fun z : TangentBundle 𝓘(ℝ,Model n) M => | |
| ((extChartAt 𝓘(ℝ,Model n) a) z.1, | |
| ((extChartAt 𝓘(ℝ,Model n) c) (riemannianExp z.1 z.2), | |
| (extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) | |
| (⟨a,0⟩ : TangentBundle 𝓘(ℝ,Model n) M) z).2))) '' | |
| {z : TangentBundle 𝓘(ℝ,Model n) M | | |
| z.2∈minimizingVectors z.1 ∧ z.1∈K ∧ riemannianExp z.1 z.2∈L} | |
| lemma isCompact_minimizingRays_between {K L : Set M} | |
| (hK : IsCompact K) (hL : IsCompact L) : | |
| IsCompact {z : TangentBundle 𝓘(ℝ,Model n) M | | |
| z.2∈minimizingVectors z.1 ∧ z.1∈K ∧ riemannianExp z.1 z.2∈L} := by | |
| exact (isCompact_total_minimizingVectors (n := n) (M := M)).inter_right | |
| ((hK.isClosed.preimage (FiberBundle.continuous_proj _ _)).inter | |
| (hL.isClosed.preimage contMDiff_riemannianExp.continuous)) | |
| lemma continuousOn_riemannianExp_chart (c : M) : | |
| ContinuousOn (fun z : TangentBundle 𝓘(ℝ,Model n) M => | |
| (extChartAt 𝓘(ℝ,Model n) c) (riemannianExp z.1 z.2)) | |
| {z | riemannianExp z.1 z.2∈(extChartAt 𝓘(ℝ,Model n) c).source} := | |
| (continuousOn_extChartAt c).comp contMDiff_riemannianExp.continuous.continuousOn | |
| (fun _ hz => hz) | |
| lemma continuousOn_endpointChartParameters {a c : M} {K L : Set M} | |
| (hKs : K⊆(extChartAt 𝓘(ℝ,Model n) a).source) | |
| (hLs : L⊆(extChartAt 𝓘(ℝ,Model n) c).source) : | |
| ContinuousOn (fun z : TangentBundle 𝓘(ℝ,Model n) M => | |
| ((extChartAt 𝓘(ℝ,Model n) a) z.1, | |
| ((extChartAt 𝓘(ℝ,Model n) c) (riemannianExp z.1 z.2), | |
| (extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) | |
| (⟨a,0⟩ : TangentBundle 𝓘(ℝ,Model n) M) z).2))) | |
| {z : TangentBundle 𝓘(ℝ,Model n) M | | |
| z.2∈minimizingVectors z.1 ∧ z.1∈K ∧ riemannianExp z.1 z.2∈L} := by | |
| let S := {z : TangentBundle 𝓘(ℝ,Model n) M | | |
| z.2∈minimizingVectors z.1 ∧ z.1∈K ∧ riemannianExp z.1 z.2∈L} | |
| have hX : ContinuousOn (extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) | |
| (⟨a,0⟩ : TangentBundle 𝓘(ℝ,Model n) M)) S := | |
| (continuousOn_extChartAt _).mono | |
| (fun z hz => (tangent_chart_source_iff _ z).mpr (hKs hz.2.1)) | |
| have hY : ContinuousOn (fun z : TangentBundle 𝓘(ℝ,Model n) M => | |
| (extChartAt 𝓘(ℝ,Model n) c) (riemannianExp z.1 z.2)) S := | |
| (continuousOn_riemannianExp_chart c).mono (fun z hz => hLs hz.2.2) | |
| have H := hX.fst.prodMk (hY.prodMk hX.snd) | |
| convert H using 1 | |
| funext z | |
| rw [tangent_chart_apply] | |
| lemma isCompact_minimizingEndpointChartParameters {a c : M} {K L : Set M} | |
| (hK : IsCompact K) (hL : IsCompact L) | |
| (hKs : K⊆(extChartAt 𝓘(ℝ,Model n) a).source) | |
| (hLs : L⊆(extChartAt 𝓘(ℝ,Model n) c).source) : | |
| IsCompact (minimizingEndpointChartParameters (n := n) a c K L) := | |
| (isCompact_minimizingRays_between hK hL).image_of_continuousOn | |
| (continuousOn_endpointChartParameters hKs hLs) | |
| omit [CompactSpace M] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| omit [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] in | |
| lemma tangent_symmL_trivialization {a x : M} | |
| (hx : x∈(extChartAt 𝓘(ℝ,Model n) a).source) | |
| (p : TangentSpace 𝓘(ℝ,Model n) x) : | |
| (trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ x | |
| ((trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).continuousLinearMapAt ℝ x p)=p := | |
| Trivialization.symmL_continuousLinearMapAt _ | |
| (by simpa only [TangentBundle.trivializationAt_baseSet,extChartAt_source] using hx) p | |
| lemma trivialization_mem_minimizingEndpointChartParameters {a c x : M} {K L : Set M} | |
| (hx : x∈K) (hxa : x∈(extChartAt 𝓘(ℝ,Model n) a).source) | |
| {p : TangentSpace 𝓘(ℝ,Model n) x} (hp : p∈minimizingVectors x) | |
| (hy : riemannianExp x p∈L) : | |
| ((extChartAt 𝓘(ℝ,Model n) a) x, | |
| ((extChartAt 𝓘(ℝ,Model n) c) (riemannianExp x p), | |
| (trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).continuousLinearMapAt ℝ x p))∈ | |
| minimizingEndpointChartParameters a c K L := by | |
| refine ⟨⟨x,p⟩,⟨hp,hx,hy⟩,?_⟩ | |
| simp only [tangent_chart_trivialization hxa] | |
| end WeakMTWTransport | |
| end | |
| end OAI | |