| import Mathlib |
| import OAI.Analysis.BiholderTransport.Regularity.ReverseRegular |
| import OAI.Analysis.BiholderTransport.Regularity.ActiveGraphLimit |
| import OAI.Analysis.BiholderTransport.Coordinates.GraphCoordinates |
| |
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
|
|
| namespace WeakMTWTransport |
| section CenterRayLimit |
| variable {n : ℕ} {M : Type*} [MetricSpace M] |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] |
|
|
| def chartRay (c:M) (b r:Model n) : TangentBundle 𝓘(ℝ,Model n) M := |
| ⟨(extChartAt 𝓘(ℝ,Model n) c).symm b,chartFiberInverse c b r⟩ |
|
|
| lemma chartRay_at_center (c:M) (r:Model n) : |
| chartRay c (extChartAt 𝓘(ℝ,Model n) c c) r=⟨c,r⟩ := by |
| simp only [chartRay,chartFiberInverse_at_center] |
| exact congrArg (fun y:M=>(⟨y,r⟩:TangentBundle 𝓘(ℝ,Model n) M)) |
| ((extChartAt 𝓘(ℝ,Model n) c).left_inv (mem_extChartAt_source c)) |
|
|
| lemma chartRay_tendsto {c:M} {b r:ℕ → Model n} {r0:Model n} |
| (hb:Tendsto b atTop (𝓝 (extChartAt 𝓘(ℝ,Model n) c c))) |
| (hr:Tendsto r atTop (𝓝 r0)) : |
| Tendsto (fun k=>chartRay c (b k) (r k)) atTop |
| (𝓝 (⟨c,r0⟩:TangentBundle 𝓘(ℝ,Model n) M)) := by |
| let χ:=extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) |
| (⟨c,0⟩:TangentBundle 𝓘(ℝ,Model n) M) |
| have hc:extChartAt 𝓘(ℝ,Model n) c c∈(extChartAt 𝓘(ℝ,Model n) c).target:= |
| (extChartAt 𝓘(ℝ,Model n) c).map_source (mem_extChartAt_source c) |
| have ht:(extChartAt 𝓘(ℝ,Model n) c c,r0)∈χ.target:= |
| (tangent_chart_target_iff _ _).mpr hc |
| have H: Tendsto (fun k=>χ.symm (b k,r k)) atTop |
| (𝓝 (χ.symm (extChartAt 𝓘(ℝ,Model n) c c,r0))):= |
| (continuousAt_extChartAt_symm'' ht).tendsto.comp (hb.prodMk_nhds hr) |
| rw [tangent_chart_symm_trivialization hc] at H |
| change Tendsto _ _ (𝓝 (chartRay c (extChartAt 𝓘(ℝ,Model n) c c) r0)) at H |
| rw [chartRay_at_center] at H |
| apply H.congr' |
| filter_upwards [hb.eventually ((isOpen_extChartAt_target _).mem_nhds hc)] with k hk |
| exact tangent_chart_symm_trivialization hk |
|
|
| section |
| variable [CompactSpace 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)] |
|
|
| lemma modified_reverseRay_tendsto {c:M} {b r:ℕ → Model n} {l:ℕ → ℝ} {r0:Model n} {l0:ℝ} |
| (hb:Tendsto b atTop (𝓝 (extChartAt 𝓘(ℝ,Model n) c c))) |
| (hr:Tendsto r atTop (𝓝 r0)) (hl:Tendsto l atTop (𝓝 l0)) : |
| Tendsto (fun k=>reverseRay (tangentScale (l k) (chartRay c (b k) (r k)))) atTop |
| (𝓝 (reverseRay (⟨c,l0 • r0⟩:TangentBundle 𝓘(ℝ,Model n) M))) := by |
| exact continuous_reverseRay.continuousAt.tendsto.comp |
| (contMDiff_tangentScale.continuous.continuousAt.tendsto.comp |
| (hl.prodMk_nhds (chartRay_tendsto hb hr))) |
|
|
| end |
|
|
| lemma graph_coordinates_tendsto {q:ℕ → TangentBundle 𝓘(ℝ,Model n) M} |
| {q0:TangentBundle 𝓘(ℝ,Model n) M} (hq:Tendsto q atTop (𝓝 q0)) : |
| Tendsto (fun k=>graphBaseCoordinate q0.1 (q k)) atTop |
| (𝓝 (extChartAt 𝓘(ℝ,Model n) q0.1 q0.1)) ∧ |
| Tendsto (fun k=>graphVelocityCoordinate q0.1 (q k)) atTop (𝓝 q0.2) := by |
| let χ:=extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) |
| (⟨q0.1,0⟩:TangentBundle 𝓘(ℝ,Model n) M) |
| have hs:q0∈χ.source:=(tangent_chart_source_iff _ _).mpr (mem_extChartAt_source _) |
| have H:=((continuousOn_extChartAt _).continuousAt |
| ((isOpen_extChartAt_source _).mem_nhds hs)).tendsto.comp hq |
| have HV:Tendsto (fun k=>graphVelocityCoordinate q0.1 (q k)) atTop |
| (𝓝 (graphVelocityCoordinate q0.1 q0)):=H.snd_nhds |
| rw [graph_coordinates_at_center] at HV |
| exact ⟨H.fst_nhds,HV⟩ |
|
|
| end CenterRayLimit |
| end WeakMTWTransport |
|
|
| end |
| end |
|
|
| end OAI |
|
|
