| import Mathlib |
| import OAI.Analysis.BiholderTransport.Geodesics.SprayScaling |
| import OAI.Analysis.BiholderTransport.Calculus.LocalMetricGradient |
| import OAI.Analysis.BiholderTransport.Coordinates.IntrinsicGradient |
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|
| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology |
| open MeasureTheory |
| open scoped ContDiff ENNReal |
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| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory |
| open scoped Topology ContDiff Manifold Bundle ENNReal |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology Manifold |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Bundle |
| open scoped Topology Bundle |
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| open scoped Topology |
| open Function Manifold Set |
| open Manifold Bundle |
| open scoped Manifold Bundle |
| open Set |
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| 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_local_cost_gradient (a : M) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ b ∈ Metric.ball a δ, ∀ y ∈ Metric.ball a δ, |
| ∃ q : TangentBundle 𝓘(ℝ,E) M, |
| q.1 = y ∧ ‖q.2‖ = dist b y ∧ (sprayFlow (-1) q).1 = b ∧ |
| HasMFDerivAt 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (fun x => dist b x ^ 2 / 2) q.1 (innerSL ℝ q.2) := by |
| obtain ⟨δ,τ,hδ,hτ,hsrc,H⟩ := exists_local_metric_gradient (E := E) a |
| refine ⟨δ,hδ,?_⟩ |
| intro b hb y hy |
| obtain ⟨z,hzb,hzy,hn,hD⟩ := H b hb y hy |
| have hzs : (sprayFlow τ z).1 ∈ (extChartAt 𝓘(ℝ,E) a).source := hzy ▸ hsrc hy |
| have hDc : HasFDerivAt (fun w : E => dist b ((extChartAt 𝓘(ℝ,E) a).symm w)^2 / 2) |
| (τ • riemannianCoordinateMetric a (extChartAt 𝓘(ℝ,E) a (sprayFlow τ z).1) |
| ((extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) |
| (⟨a,0⟩ : TangentBundle 𝓘(ℝ,E) M) (sprayFlow τ z)).2)) |
| (extChartAt 𝓘(ℝ,E) a (sprayFlow τ z).1) := by rwa [hzy] |
| have hD' := hasMFDerivAt_of_coordinate_gradient (sprayFlow τ z) hzs |
| (fun x => dist b x^2/2) τ hDc |
| refine ⟨tangentScale τ (sprayFlow τ z),hzy,?_,?_,?_⟩ |
| · change ‖τ • (sprayFlow τ z).2‖ = _ |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos hτ,sprayFlow_speed] |
| exact hn.symm |
| · rw [sprayFlow_scale,show τ * (-1) = -τ by ring,←sprayFlow_add,neg_add_cancel,sprayFlow_zero] |
| exact hzb |
| · exact hD'.congr_mfderiv (map_smul (innerSL ℝ) τ (sprayFlow τ z).2).symm |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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