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110f4f3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import Mathlib
import OAI.Analysis.BiholderTransport.Geodesics.SprayScaling
import OAI.Analysis.BiholderTransport.Calculus.LocalMetricGradient
import OAI.Analysis.BiholderTransport.Coordinates.IntrinsicGradient
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_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
end WeakMTWTransport
end
end OAI
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