| import Mathlib |
| import OAI.Analysis.BiholderTransport.Coordinates.JointNormalChart |
| import OAI.Analysis.BiholderTransport.Geodesics.EndpointIdentity |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
|
|
| open Set Filter |
| open scoped Topology NNReal |
|
|
| open Set Filter |
| open scoped Topology |
|
|
| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
|
| open Set |
| open scoped Topology |
|
|
| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
|
|
| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
|
|
| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
|
|
| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
|
|
| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
|
|
| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
|
|
| open Set Filter ContinuousLinearMap |
| open scoped Topology |
| open MeasureTheory |
| open scoped ContDiff ENNReal |
|
|
| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory |
| open scoped Topology ContDiff Manifold Bundle ENNReal |
|
|
| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology Manifold |
|
|
| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
|
|
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
|
|
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
|
|
| open Set Filter Bundle |
| open scoped Topology Bundle |
|
|
| open scoped Topology |
| open Function Manifold Set |
| open Manifold Bundle |
| open scoped Manifold Bundle |
| open Set |
|
|
| namespace WeakMTWTransport |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] |
| {M : Type*} [MetricSpace M] [CompactSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,E) x)] |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,E) ∞ E (fun x : M => TangentSpace 𝓘(ℝ,E) x)] |
| [IsRiemannianManifold 𝓘(ℝ,E) M] |
|
|
| lemma exists_metric_normal_flow (a : M) : |
| ∃ δ r τ : ℝ, 0 < δ ∧ 0 < r ∧ 0 < τ ∧ τ < r ∧ |
| ∃ Ψ W : ℝ × E → E, ∃ e : OpenPartialHomeomorph E E, |
| ContDiffOn ℝ ∞ Ψ (Ioo (-r) r ×ˢ Metric.ball 0 r) ∧ |
| ContDiffOn ℝ ∞ W (Ioo (-r) r ×ˢ Metric.ball 0 r) ∧ |
| (∀ v ∈ Metric.ball (0:E) r, Ψ (0,v) = extChartAt 𝓘(ℝ,E) a a ∧ W (0,v) = v) ∧ |
| (∀ t ∈ Ioo (-r) r, ∀ v ∈ Metric.ball (0:E) r, |
| Ψ (t,v) ∈ (extChartAt 𝓘(ℝ,E) a).target ∧ |
| HasDerivAt (fun s => Ψ (s,v)) (W (t,v)) t ∧ |
| HasDerivAt (fun s => W (s,v)) |
| (-coordinateChristoffel (riemannianCoordinateMetric a) (Ψ (t,v)) (W (t,v)) (W (t,v))) t) ∧ |
| (e : E → E) = (fun v => Ψ (τ,v)) ∧ |
| 0 ∈ e.source ∧ e 0 = extChartAt 𝓘(ℝ,E) a a ∧ e.source ⊆ Metric.ball (0:E) r ∧ |
| ContDiffOn ℝ ∞ e e.source ∧ ContDiffOn ℝ ∞ e.symm e.target ∧ |
| (Metric.ball a δ ⊆ (extChartAt 𝓘(ℝ,E) a).source ∩ |
| (extChartAt 𝓘(ℝ,E) a) ⁻¹' e.target) ∧ |
| (∀ y ∈ Metric.ball a δ, dist a y = τ * |
| Real.sqrt (riemannianCoordinateMetric a (extChartAt 𝓘(ℝ,E) a a) |
| (e.symm (extChartAt 𝓘(ℝ,E) a y)) (e.symm (extChartAt 𝓘(ℝ,E) a y)))) ∧ |
| fderiv ℝ e 0 = τ • ContinuousLinearMap.id ℝ E := by |
| obtain ⟨δ,r,τ,hδ,hr,hτ,hτr,Φ,e,hΦ,hΦ0,hode,he,hesub,hed,hei,hsrc,hdiag,htarget,hdist⟩ := |
| exists_uniform_metric_normal_coordinates (E := E) a |
| let c := extChartAt 𝓘(ℝ,E) a |
| let x := c a |
| let Ψ : ℝ × E → E := fun z => (Φ (z.1,(x,z.2))).1 |
| let W : ℝ × E → E := fun z => (Φ (z.1,(x,z.2))).2 |
| have hball : ∀ v ∈ Metric.ball (0:E) r, (x,v) ∈ Metric.ball (x,0) r := by |
| intro v hv |
| simpa only [Metric.mem_ball,Prod.dist_eq,dist_self,max_lt_iff] using And.intro hr hv |
| have hΦc := hΦ.comp (contDiffOn_fst.prodMk (contDiffOn_const.prodMk contDiffOn_snd)) |
| (show MapsTo (fun z : ℝ × E => (z.1,(x,z.2))) |
| (Ioo (-r) r ×ˢ Metric.ball 0 r) (Ioo (-r) r ×ˢ Metric.ball (x,0) r) from |
| fun z hz => ⟨hz.1,hball z.2 hz.2⟩) |
| have hp : ∀ z : E × E, (e z).1 = z.1 := by intro z; rw [he] |
| let f := fiberwiseHomeomorph e hp x |
| have hf : (f : E → E) = fun v => Ψ (τ,v) := by funext v; change (e (x,v)).2 = _; rw [he] |
| have h0 : (0:ℝ) ∈ Ioo (-r) r := ⟨by linarith,hr⟩ |
| have ht : τ ∈ Ioo (-r) r := ⟨by linarith,hτr⟩ |
| have haz : a ∈ Metric.ball a δ := Metric.mem_ball_self hδ |
| have hzero := coordinate_geodesic_zero_velocity |
| (fun t ht' => ((contDiffOn_riemannianCoordinateMetric a).contDiffAt |
| ((isOpen_extChartAt_target a).mem_nhds |
| (hode t ht' (x,0) (Metric.mem_ball_self hr)).1)).differentiableAt (by simp)) |
| (fun t ht' => riemannianCoordinateMetric_isInvertible |
| (hode t ht' (x,0) (Metric.mem_ball_self hr)).1) |
| (fun t ht' _ hv => riemannianCoordinateMetric_positive |
| (hode t ht' (x,0) (Metric.mem_ball_self hr)).1 hv) |
| (fun t _ u v => riemannianCoordinateMetric_symm a _ u v) |
| (fun t ht' => ⟨((hode t ht' (x,0) (Metric.mem_ball_self hr)).2).fst, |
| ((hode t ht' (x,0) (Metric.mem_ball_self hr)).2).snd⟩) h0 |
| (show (Φ (0,(x,0))).2 = 0 from congrArg Prod.snd (hΦ0 (x,0) (Metric.mem_ball_self hr))) ht |
| have hf0 : f 0 = x := by |
| rw [hf] |
| exact hzero.1.trans (congrArg Prod.fst (hΦ0 (x,0) (Metric.mem_ball_self hr))) |
| refine ⟨δ,r,τ,hδ,hr,hτ,hτr,Ψ,W,f,hΦc.fst,hΦc.snd,?_,?_,hf,hdiag a haz,hf0,?_, |
| fiberwiseHomeomorph_contDiffOn hp x hed,fiberwiseHomeomorph_symm_contDiffOn hp x hei,?_,?_,?_⟩ |
| · intro v hv |
| exact ⟨congrArg Prod.fst (hΦ0 (x,v) (hball v hv)),congrArg Prod.snd (hΦ0 (x,v) (hball v hv))⟩ |
| · intro t ht' v hv |
| exact ⟨(hode t ht' (x,v) (hball v hv)).1, |
| ((hode t ht' (x,v) (hball v hv)).2).fst,((hode t ht' (x,v) (hball v hv)).2).snd⟩ |
| · intro v hv |
| exact (max_lt_iff.mp (show max (dist x x) (dist v 0) < r from hesub hv)).2 |
| · intro y hy |
| exact ⟨hsrc hy,htarget a haz y hy⟩ |
| · intro y hy |
| apply hdist a haz y hy (f.symm (c y)) |
| · exact f.map_target (htarget a haz y hy) |
| · change Ψ (τ,f.symm (c y)) = c y |
| exact (congrFun hf _).symm.trans (f.right_inv (htarget a haz y hy)) |
| · rw [hf] |
| exact coordinate_flow_endpoint_derivative_zero a hr hΦ hΦ0 hode ht |
|
|
| end WeakMTWTransport |
|
|
| end |
|
|
| end OAI |
|
|
