| import Mathlib |
| import OAI.Analysis.BiholderTransport.Geodesics.NoCorner |
| import OAI.Analysis.BiholderTransport.Geodesics.IntegratedGauss |
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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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| open Set Filter |
| open scoped Topology ContDiff |
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| namespace WeakMTWTransport |
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| section |
| 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] |
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| lemma exists_local_spray_cost_gradient (a : M) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ z : TangentBundle 𝓘(ℝ,E) M, ∀ t : ℝ, 0 < t → |
| z.1 ∈ Metric.ball a δ → (sprayFlow t z).1 ∈ Metric.ball a δ → |
| dist z.1 (sprayFlow t z).1 = t*‖z.2‖ → |
| HasMFDerivAt 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (fun y => dist z.1 y^2/2) |
| (sprayFlow t z).1 (innerSL ℝ (t • (sprayFlow t z).2)) := by |
| obtain ⟨δ,hδ,H⟩ := exists_local_cost_gradient (E := E) a |
| refine ⟨δ,hδ,?_⟩ |
| intro z t ht hz hy hmin |
| obtain ⟨⟨b,w⟩,hb,hn,_,hD⟩ := H z.1 hz (sprayFlow t z).1 hy |
| dsimp only at hb |
| subst b |
| have hw := sprayFlow_minimizer_gradient z ht hmin w hn hD |
| rwa [hw] at hD |
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| omit [IsRiemannianManifold 𝓘(ℝ,E) M] in |
| lemma sprayFlow_reverse (z : TangentBundle 𝓘(ℝ,E) M) (t : ℝ) : |
| sprayFlow t (tangentScale (-1) (sprayFlow t z)) = tangentScale (-1) z := by |
| rw [sprayFlow_scale,neg_one_mul,←sprayFlow_add,neg_add_cancel,sprayFlow_zero] |
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| lemma exists_local_spray_cost_start_gradient (a : M) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ z : TangentBundle 𝓘(ℝ,E) M, ∀ t : ℝ, 0 < t → |
| z.1 ∈ Metric.ball a δ → (sprayFlow t z).1 ∈ Metric.ball a δ → |
| dist z.1 (sprayFlow t z).1 = t*‖z.2‖ → |
| HasMFDerivAt 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (fun y => dist y (sprayFlow t z).1^2/2) |
| z.1 (innerSL ℝ ((-t) • z.2)) := by |
| obtain ⟨δ,hδ,H⟩ := exists_local_spray_cost_gradient (E := E) a |
| refine ⟨δ,hδ,?_⟩ |
| intro z t ht hz hy hmin |
| let w := tangentScale (-1) (sprayFlow t z) |
| have hw : sprayFlow t w = tangentScale (-1) z := sprayFlow_reverse z t |
| have hb : (sprayFlow t w).1 ∈ Metric.ball a δ := by rw [hw]; exact hz |
| have hm : dist w.1 (sprayFlow t w).1 = t*‖w.2‖ := by |
| rw [hw] |
| change dist (sprayFlow t z).1 z.1 = t*‖(-1:ℝ) • (sprayFlow t z).2‖ |
| rw [neg_one_smul,norm_neg,sprayFlow_speed,dist_comm,hmin] |
| have hd := H w t ht hy hb hm |
| rw [hw] at hd |
| change HasMFDerivAt 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (fun y => dist (sprayFlow t z).1 y^2/2) |
| z.1 (innerSL ℝ (t • ((-1:ℝ) • z.2))) at hd |
| rw [smul_smul,mul_neg_one] at hd |
| have hf : (fun y => dist y (sprayFlow t z).1^2/2) = |
| (fun y => dist (sprayFlow t z).1 y^2/2) := by |
| funext y; rw [dist_comm] |
| rw [hf] |
| exact hd |
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| end |
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| 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] |
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| omit [FiniteDimensional ℝ E] [CompactSpace M] |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,E) ∞ E (fun x : M => TangentSpace 𝓘(ℝ,E) x)] |
| [IsRiemannianManifold 𝓘(ℝ,E) M] in |
| lemma contMDiff_fiber_embedding (x : M) : |
| ContMDiff 𝓘(ℝ,TangentSpace 𝓘(ℝ,E) x) (𝓘(ℝ,E).prod 𝓘(ℝ,E)) ∞ |
| (fun v : TangentSpace 𝓘(ℝ,E) x => (⟨x,v⟩ : TangentBundle 𝓘(ℝ,E) M)) := by |
| intro v |
| rw [Bundle.contMDiffAt_totalSpace] |
| refine ⟨contMDiffAt_const,?_⟩ |
| let e := trivializationAt E (fun y : M => TangentSpace 𝓘(ℝ,E) y) x |
| let L : TangentSpace 𝓘(ℝ,E) x →L[ℝ] E := e.continuousLinearMapAt ℝ x |
| have heq : (fun w : TangentSpace 𝓘(ℝ,E) x => (e (⟨x,w⟩ : TangentBundle 𝓘(ℝ,E) M)).2) = L := by |
| funext w |
| exact (e.continuousLinearMapAt_apply_of_mem ℝ (mem_baseSet_trivializationAt E _ x) _).symm |
| change ContMDiffAt 𝓘(ℝ,TangentSpace 𝓘(ℝ,E) x) 𝓘(ℝ,E) ∞ |
| (fun w => (e ⟨x,w⟩).2) v |
| rw [heq] |
| exact L.contDiff.contDiffAt.contMDiffAt |
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| omit [IsRiemannianManifold 𝓘(ℝ,E) M] in |
| lemma contMDiff_spray_fiber (x : M) (t : ℝ) : |
| ContMDiff 𝓘(ℝ,TangentSpace 𝓘(ℝ,E) x) (𝓘(ℝ,E).prod 𝓘(ℝ,E)) ∞ |
| (fun v : TangentSpace 𝓘(ℝ,E) x => sprayFlow t (⟨x,v⟩ : TangentBundle 𝓘(ℝ,E) M)) := |
| contMDiff_sprayFlow.comp (contMDiff_const.prodMk (contMDiff_fiber_embedding x)) |
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| omit [IsRiemannianManifold 𝓘(ℝ,E) M] in |
| lemma spray_action_line_stationary (x : M) (p w : TangentSpace 𝓘(ℝ,E) x) |
| (L ε : ℝ) (hε : ε ≠ 0) (y : M) |
| (hD : HasMFDerivAt 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (fun b => dist b y^2/2) |
| (sprayFlow L (⟨x,p⟩ : TangentBundle 𝓘(ℝ,E) M)).1 |
| (innerSL ℝ ((-ε) • (sprayFlow L (⟨x,p⟩ : TangentBundle 𝓘(ℝ,E) M)).2))) : |
| HasDerivAt (fun s : ℝ => L*‖p+s • w‖^2/2 + |
| (dist (sprayFlow L (⟨x,p+s • w⟩ : TangentBundle 𝓘(ℝ,E) M)).1 y^2/2)/ε) 0 0 := by |
| let ν : ℝ → TangentBundle 𝓘(ℝ,E) M := fun s => sprayFlow L ⟨x,p+s • w⟩ |
| let ψ : ℝ → M := fun s => (ν s).1 |
| have hψ : ContMDiff 𝓘(ℝ,ℝ) 𝓘(ℝ,E) ∞ ψ := |
| (Bundle.contMDiff_proj (fun b : M => TangentSpace 𝓘(ℝ,E) b)).comp |
| (contMDiff_sprayFlow.comp (contMDiff_const.prodMk (contMDiff_fiber_line x p w))) |
| have hν0 : ν 0 = sprayFlow L (⟨x,p⟩ : TangentBundle 𝓘(ℝ,E) M) := by simp [ν] |
| have hD₀ : HasMFDerivAt 𝓘(ℝ,E) 𝓘(ℝ,ℝ) (fun b => dist b y^2/2) (ν 0).1 |
| (innerSL ℝ ((-ε) • (ν 0).2)) := by rw [hν0]; exact hD |
| have hDc := hD₀.comp 0 (hψ.mdifferentiable (by simp) 0).hasMFDerivAt |
| let J := mfderiv 𝓘(ℝ,ℝ) 𝓘(ℝ,E) ψ 0 (1:ℝ) |
| have hd : HasDerivAt (fun s => dist (ψ s) y^2/2) |
| ((-ε) * inner ℝ (ν 0).2 J) 0 := by |
| have hf : HasFDerivAt (fun s => dist (ψ s) y^2/2) |
| ((innerSL ℝ ((-ε) • (ν 0).2)).comp (mfderiv 𝓘(ℝ,ℝ) 𝓘(ℝ,E) ψ 0)) 0 := by |
| convert! hDc.hasFDerivAt using 1 |
| convert! hf.hasDerivAt using 1 |
| change (-ε) * inner ℝ (ν 0).2 J = inner ℝ ((-ε) • (ν 0).2) J |
| rw [real_inner_smul_left] |
| have hg : inner ℝ (ν 0).2 J = L*inner ℝ p w := by |
| have he : HasDerivAt (fun s : ℝ => ‖p+s • w‖^2) (2*inner ℝ p w) 0 := by |
| convert! ((hasDerivAt_const (0:ℝ) p).add ((hasDerivAt_id (0:ℝ)).smul_const w)).norm_sq using 1 |
| simp |
| have H := spray_variation_pairing_affine (ν := fun s => (⟨x,p+s • w⟩ : TangentBundle 𝓘(ℝ,E) M)) |
| (contMDiff_fiber_line x p w) he L |
| convert! H using 1 |
| simp |
| have hn := ((hasDerivAt_const (0:ℝ) p).add ((hasDerivAt_id (0:ℝ)).smul_const w)).norm_sq |
| have he : HasDerivAt (fun s : ℝ => L*‖p+s • w‖^2/2) (L*inner ℝ p w) 0 := by |
| convert! (hn.const_mul L).div_const 2 using 1 |
| simp |
| ring |
| have H := he.add (hd.div_const ε) |
| apply H.congr_deriv |
| rw [hg] |
| field_simp |
| ring |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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