| import Mathlib |
| import OAI.Analysis.BiholderTransport.Coordinates.JointNormalChart |
|
|
| 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] [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_local_cost_smooth (a : M) : |
| ∃ δ : ℝ, 0<δ ∧ ContMDiffOn (𝓘(ℝ,E).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ |
| (fun z : M×M => dist z.1 z.2^2/2) (Metric.ball a δ ×ˢ Metric.ball a δ) := by |
| obtain ⟨δ,_,τ,hδ,_,_,_,Φ,e,_,_,_,he,_,_,hei,hsrc,_,htarget,hdist⟩ := |
| exists_uniform_metric_normal_coordinates (E := E) a |
| let c := extChartAt 𝓘(ℝ,E) a |
| let g := riemannianCoordinateMetric (E := E) a |
| let V : M×M → E := fun z => (e.symm (c z.1,c z.2)).2 |
| let f : M×M → ℝ := fun z => (τ^2/2)*g (c z.1) (V z) (V z) |
| have heq : ∀ z ∈ Metric.ball a δ ×ˢ Metric.ball a δ, dist z.1 z.2^2/2 = f z := by |
| rintro ⟨b,y⟩ ⟨hb,hy⟩ |
| have ht := htarget b hb y hy |
| have hs := e.map_target ht |
| have hi := e.right_inv ht |
| have hfst : (e.symm (c b,c y)).1=c b := by |
| have hh := congrArg Prod.fst hi |
| rw [he] at hh |
| exact hh |
| have hpair : e.symm (c b,c y)=(c b,V (b,y)) := Prod.ext hfst rfl |
| rw [hpair] at hs hi |
| have hval : (Φ (τ,(c b,V (b,y)))).1=c y := by |
| have hh := congrArg Prod.snd hi |
| rw [he] at hh |
| exact hh |
| have hd := hdist b hb y hy (V (b,y)) hs hval |
| have hnonneg : 0 ≤ g (c b) (V (b,y)) (V (b,y)) := by |
| by_cases hv : V (b,y)=0 |
| · simp [hv] |
| · exact (riemannianCoordinateMetric_positive (c.map_source (hsrc hb)) hv).le |
| change dist b y^2/2 = (τ^2/2)*g (c b) (V (b,y)) (V (b,y)) |
| rw [hd,mul_pow,Real.sq_sqrt hnonneg] |
| ring |
| refine ⟨δ,hδ,?_⟩ |
| rintro z hz |
| have hc₁ : ContMDiffAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ (fun z : M×M => c z.1) z := |
| (contMDiffAt_extChartAt' (I := 𝓘(ℝ,E)) (by simpa only [extChartAt_source] using hsrc hz.1)).comp z contMDiffAt_fst |
| have hc₂ : ContMDiffAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ (fun z : M×M => c z.2) z := |
| (contMDiffAt_extChartAt' (I := 𝓘(ℝ,E)) (by simpa only [extChartAt_source] using hsrc hz.2)).comp z contMDiffAt_snd |
| have heSnd : ContDiffAt ℝ ∞ (fun w : E×E => (e.symm w).2) (c z.1,c z.2) := |
| (hei.contDiffAt (e.open_target.mem_nhds (htarget z.1 hz.1 z.2 hz.2))).snd |
| have heSndM : ContMDiffAt 𝓘(ℝ,E×E) 𝓘(ℝ,E) ∞ |
| (fun w : E×E => (e.symm w).2) (c z.1,c z.2) := heSnd.contMDiffAt |
| have hV : ContMDiffAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) 𝓘(ℝ,E) ∞ V z := |
| heSndM.comp (f := fun z : M×M => (c z.1,c z.2)) z (hc₁.prodMk_space hc₂) |
| have hg : ContMDiffAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) 𝓘(ℝ,E →L[ℝ] E →L[ℝ] ℝ) ∞ |
| (fun z : M×M => g (c z.1)) z := |
| ((contDiffOn_riemannianCoordinateMetric a).contDiffAt |
| ((isOpen_extChartAt_target a).mem_nhds (c.map_source (hsrc hz.1)))).contMDiffAt.comp z hc₁ |
| have hf : ContMDiffAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) 𝓘(ℝ,ℝ) ∞ f z := |
| contMDiffAt_const.mul ((hg.clm_apply hV).clm_apply hV) |
| exact ((hf.congr_of_eventuallyEq (Filter.eventuallyEq_of_mem |
| ((Metric.isOpen_ball.prod Metric.isOpen_ball).mem_nhds hz) heq)).contMDiffWithinAt) |
|
|
| end WeakMTWTransport |
|
|
| end |
|
|
| end OAI |
|
|
