openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /Geodesics /MinimizingChartParameters.lean
| 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 | |