| import Mathlib |
| import OAI.Analysis.BiholderTransport.Model |
| import OAI.Analysis.BiholderTransport.Geodesics.SprayMinimizing |
| import OAI.Analysis.BiholderTransport.Geodesics.EndpointIdentity |
|
|
| 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 |
|
|
| section |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] |
|
|
| lemma sprayFlow_isGeodesicWithInitialData (x : M) (v : TangentSpace 𝓘(ℝ,Model n) x) : |
| IsGeodesicWithInitialData x v (fun t => (sprayFlow t ⟨x,v⟩).1) := by |
| have hsm : ContMDiff 𝓘(ℝ,ℝ) (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) ∞ |
| (fun t => sprayFlow t (⟨x,v⟩ : TangentBundle 𝓘(ℝ,Model n) M)) := |
| contMDiff_sprayFlow.comp (contMDiff_id.prodMk contMDiff_const) |
| refine ⟨(Bundle.contMDiff_proj (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)).comp hsm,?_,?_⟩ |
| · rw [spray_velocity_eq_mfderiv ((sprayFlow_curve ⟨x,v⟩).isMIntegralCurveAt 0)] |
| exact sprayFlow_zero ⟨x,v⟩ |
| · exact sprayFlow_locally_minimizing ⟨x,v⟩ |
|
|
| lemma exists_geodesicWithInitialData (x : M) (v : TangentSpace 𝓘(ℝ,Model n) x) : |
| ∃ γ : ℝ → M, IsGeodesicWithInitialData x v γ := |
| ⟨_,sprayFlow_isGeodesicWithInitialData x v⟩ |
|
|
| lemma riemannianExp_eq_chosen_geodesic (x : M) (v : TangentSpace 𝓘(ℝ,Model n) x) : |
| riemannianExp x v = Classical.choose (exists_geodesicWithInitialData x v) 1 := by |
| rw [riemannianExp,dite_eq_left (exists_geodesicWithInitialData x v)] |
|
|
| end |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] |
| {M : Type*} [TopologicalSpace M] [T2Space 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)] |
|
|
| lemma coordinate_flow_eq_spray (a : M) {r : ℝ} (hr : 0 < r) |
| {Ψ W : ℝ × E → E} |
| (hinit : ∀ v ∈ Metric.ball (0:E) r, |
| Ψ (0,v) = extChartAt 𝓘(ℝ,E) a a ∧ W (0,v) = v) |
| (hode : ∀ 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) |
| {v : E} (hv : v ∈ Metric.ball (0:E) r) {t : ℝ} (ht : t ∈ Ioo (-r) r) : |
| Ψ (t,v) = extChartAt 𝓘(ℝ,E) a (sprayFlow t ⟨a,v⟩).1 := by |
| let z : TangentBundle 𝓘(ℝ,E) M := ⟨a,0⟩ |
| have H := sprayFlow_eq_coordinate_curve z hr |
| (fun s hs => (hode s hs v hv).1) |
| (fun s hs => (hode s hs v hv).2.1.prodMk (hode s hs v hv).2.2) |
| (show (extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) z).symm (Ψ (0,v), W (0,v)) = ⟨a,v⟩ by |
| rw [(hinit v hv).1,(hinit v hv).2] |
| exact tangent_chart_at_self_base_symm z v) ht |
| have Hc := congrArg (fun p : TangentBundle 𝓘(ℝ,E) M => extChartAt 𝓘(ℝ,E) a p.1) H |
| rw [tangent_chart_symm_base,(extChartAt 𝓘(ℝ,E) a).right_inv (hode t ht v hv).1] at Hc |
| exact Hc.symm |
|
|
| lemma coordinate_endpoint_radial (a : M) {r : ℝ} (hr : 0 < r) |
| {Ψ W : ℝ × E → E} |
| (hΨ : ContDiffOn ℝ ∞ Ψ (Ioo (-r) r ×ˢ Metric.ball 0 r)) |
| (hinit : ∀ v ∈ Metric.ball (0:E) r, |
| Ψ (0,v) = extChartAt 𝓘(ℝ,E) a a ∧ W (0,v) = v) |
| (hode : ∀ 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) |
| {v : E} (hv : v ∈ Metric.ball (0:E) r) {τ : ℝ} (hτ : τ ∈ Ioo (-r) r) : |
| fderiv ℝ (fun w => Ψ (τ,w)) v v = τ • W (τ,v) := by |
| have hn1 : ∀ᶠ s : ℝ in 𝓝 1, s • v ∈ Metric.ball (0:E) r := |
| ((continuousAt_id.smul continuousAt_const) : ContinuousAt (fun s : ℝ => s • v) 1) |
| (by change Metric.ball (0:E) r ∈ 𝓝 ((1:ℝ) • v) |
| simpa only [one_smul] using Metric.isOpen_ball.mem_nhds hv) |
| have hn2 : ∀ᶠ s : ℝ in 𝓝 1, s*τ ∈ Ioo (-r) r := |
| ((continuousAt_id.mul continuousAt_const) : |
| ContinuousAt (fun scalar : ℝ => scalar * τ) 1).preimage_mem_nhds |
| (by change Ioo (-r) r ∈ 𝓝 ((1:ℝ)*τ) |
| simpa only [one_mul] using isOpen_Ioo.mem_nhds hτ) |
| have heq : (fun s : ℝ => Ψ (τ,s • v)) =ᶠ[𝓝 1] (fun s => Ψ (s*τ,v)) := by |
| filter_upwards [hn1,hn2] with s hs ht |
| rw [coordinate_flow_eq_spray a hr hinit hode hs hτ, |
| coordinate_flow_eq_spray a hr hinit hode hv ht] |
| rw [show (⟨a,s • v⟩ : TangentBundle 𝓘(ℝ,E) M) = tangentScale s ⟨a,v⟩ from rfl, |
| sprayFlow_scale] |
| rfl |
| have hD : DifferentiableAt ℝ (fun w => Ψ (τ,w)) v := |
| ((hΨ.contDiffAt ((isOpen_Ioo.prod Metric.isOpen_ball).mem_nhds ⟨hτ,hv⟩)).comp v |
| (contDiffAt_const.prodMk contDiffAt_id)).differentiableAt (by simp) |
| have hleft : HasDerivAt (fun s : ℝ => Ψ (τ,s • v)) |
| (fderiv ℝ (fun w => Ψ (τ,w)) v v) 1 := by |
| have hs : HasDerivAt (fun s : ℝ => s • v) v 1 := by |
| simpa only [one_smul] using (hasDerivAt_id' (1:ℝ)).smul_const v |
| have hD' : HasFDerivAt (fun w => Ψ (τ,w)) |
| (fderiv ℝ (fun w => Ψ (τ,w)) v) ((1:ℝ) • v) := by |
| simpa only [one_smul] using hD.hasFDerivAt |
| exact hD'.comp_hasDerivAt 1 hs |
| have hright : HasDerivAt (fun s : ℝ => Ψ (s*τ,v)) (τ • W (τ,v)) 1 := by |
| have hs : HasDerivAt (fun s : ℝ => s*τ) τ 1 := hasDerivAt_mul_const τ |
| have hW : HasDerivAt (fun s => Ψ (s,v)) (W (τ,v)) ((1:ℝ)*τ) := by |
| simpa only [one_mul] using (hode τ hτ v hv).2.1 |
| exact hW.scomp 1 hs |
| exact (hleft.congr_of_eventuallyEq heq.symm).unique hright |
| |
| end WeakMTWTransport |
| |
| end |
| |
| end OAI |
| |
