| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Coordinates.ChartGradientMap | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty 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] | |
| abbrev Phase (n:ℕ) := Model n × (Model n →L[ℝ] ℝ) | |
| def phaseRaise (a:M) (q:Phase n) : Model n × Model n := | |
| (q.1,(riemannianCoordinateMetric a q.1).inverse q.2) | |
| def phaseLower (a:M) (q:Model n × Model n) : Phase n := | |
| (q.1,riemannianCoordinateMetric a q.1 q.2) | |
| def coordinatePhaseFlow (a:M) (q:ℝ×Phase n) : Phase n := | |
| let χ := extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) | |
| (⟨a,0⟩:TangentBundle 𝓘(ℝ,Model n) M) | |
| phaseLower a (χ (sprayFlow (-q.1) (χ.symm (phaseRaise a q.2)))) | |
| omit [CompactSpace M] [Nonempty M] [IsRiemannianManifold 𝓘(ℝ,Model n) M] in | |
| lemma phaseLower_contDiffAt {a:M} {q:Model n × Model n} | |
| (hq:q.1∈(extChartAt 𝓘(ℝ,Model n) a).target) : | |
| ContDiffAt ℝ ∞ (phaseLower a) q := by | |
| have hg := (contDiffOn_riemannianCoordinateMetric (E := Model n) a).contDiffAt | |
| ((isOpen_extChartAt_target a).mem_nhds hq) | |
| exact contDiffAt_fst.prodMk ((hg.comp q contDiffAt_fst).clm_apply contDiffAt_snd) | |
| omit [CompactSpace M] [Nonempty 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 phaseLower_raise {a:M} {q:Phase n} | |
| (hq:q.1∈(extChartAt 𝓘(ℝ,Model n) a).target) : phaseLower a (phaseRaise a q)=q := by | |
| exact Prod.ext rfl ((riemannianCoordinateMetric_isInvertible hq).self_apply_inverse q.2) | |
| omit [Nonempty M] [IsRiemannianManifold 𝓘(ℝ,Model n) M] in | |
| lemma coordinatePhaseFlow_zero {a:M} {q:Phase n} | |
| (hq:q.1∈(extChartAt 𝓘(ℝ,Model n) a).target) : coordinatePhaseFlow a (0,q)=q := by | |
| let χ := extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) | |
| (⟨a,0⟩:TangentBundle 𝓘(ℝ,Model n) M) | |
| have ht:phaseRaise a q∈χ.target := (tangent_chart_target_iff _ _).mpr hq | |
| dsimp only [coordinatePhaseFlow] | |
| rw [neg_zero,sprayFlow_zero,χ.right_inv ht] | |
| exact phaseLower_raise hq | |
| omit [Nonempty M] [IsRiemannianManifold 𝓘(ℝ,Model n) M] in | |
| lemma coordinatePhaseFlow_contDiffAt {a:M} {q:Phase n} | |
| (hq:q.1∈(extChartAt 𝓘(ℝ,Model n) a).target) : | |
| ContDiffAt ℝ ∞ (coordinatePhaseFlow a) (0,q) := by | |
| let χ := extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) | |
| (⟨a,0⟩:TangentBundle 𝓘(ℝ,Model n) M) | |
| have ht:phaseRaise a q∈χ.target := (tangent_chart_target_iff _ _).mpr hq | |
| have hi := (contMDiffOn_extChartAt_symm (n := ∞) (⟨a,0⟩:TangentBundle 𝓘(ℝ,Model n) M)).contMDiffAt | |
| ((isOpen_extChartAt_target _).mem_nhds ht) | |
| have hr:ContDiffAt ℝ ∞ (phaseRaise a) q := coordinateRaise_contDiffAt hq | |
| have hinput := hi.comp (f:=fun p:ℝ×Phase n=>phaseRaise a p.2) (0,q) | |
| (hr.comp (0,q) contDiffAt_snd).contMDiffAt | |
| have hf := contMDiff_sprayFlow.contMDiffAt.comp (f:=fun p:ℝ×Phase n=> | |
| (-p.1,χ.symm (phaseRaise a p.2))) (0,q) | |
| (contDiffAt_fst.neg.contMDiffAt.prodMk hinput) | |
| have he:sprayFlow (-(0:ℝ)) (χ.symm (phaseRaise a q))=χ.symm (phaseRaise a q) := by | |
| rw [neg_zero,sprayFlow_zero] | |
| have hc:ContMDiffAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) | |
| 𝓘(ℝ,Model n × Model n) ∞ χ (sprayFlow (-(0:ℝ)) (χ.symm (phaseRaise a q))) := by | |
| rw [he] | |
| exact contMDiffAt_extChartAt' (by simpa only [χ,extChartAt_source] using χ.map_target ht) | |
| have hout := (hc.comp (f:=fun p:ℝ×Phase n=> | |
| sprayFlow (-p.1) (χ.symm (phaseRaise a p.2))) (0,q) hf).contDiffAt | |
| have hbase:(χ (sprayFlow (-(0:ℝ)) (χ.symm (phaseRaise a q)))).1∈ | |
| (extChartAt 𝓘(ℝ,Model n) a).target := by | |
| rw [he,χ.right_inv ht]; exact hq | |
| exact (phaseLower_contDiffAt hbase).comp (0,q) hout | |
| omit [Nonempty M] [IsRiemannianManifold 𝓘(ℝ,Model n) M] in | |
| lemma coordinatePhaseFlow_derivative_limit {a:M} {q:Phase n} | |
| (hq:q.1∈(extChartAt 𝓘(ℝ,Model n) a).target) : | |
| Tendsto (fun p:ℝ×Phase n=>fderiv ℝ (fun v=>coordinatePhaseFlow a (p.1,v)) p.2) | |
| (𝓝 (0,q)) (𝓝 (ContinuousLinearMap.id ℝ (Phase n))) := by | |
| have hd := ContDiffAt.partial_snd_fderiv (f:=fun t v=>coordinatePhaseFlow a (t,v)) (coordinatePhaseFlow_contDiffAt hq) | |
| have he:(fun v:Phase n=>coordinatePhaseFlow a (0,v))=ᶠ[𝓝 q] id := by | |
| filter_upwards [continuousAt_fst.preimage_mem_nhds ((isOpen_extChartAt_target a).mem_nhds hq)] with v hv | |
| exact coordinatePhaseFlow_zero hv | |
| have hev:fderiv ℝ (fun v=>coordinatePhaseFlow a (0,v)) q=ContinuousLinearMap.id ℝ (Phase n) := by | |
| rw [he.fderiv_eq,fderiv_id] | |
| simpa only [hev] using hd.continuousAt.tendsto | |
| end WeakMTWTransport | |
| end | |
| end OAI | |