| import Mathlib |
| import OAI.Analysis.BiholderTransport.Coordinates.CoordinateVariation |
| import OAI.Analysis.BiholderTransport.Geodesics.CompleteSpray |
| import OAI.Analysis.BiholderTransport.LocalFlow.SmoothIntegralCurve |
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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 |
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| namespace WeakMTWTransport |
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| section |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] |
| {M : Type*} [TopologicalSpace 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)] |
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| lemma hasMFDerivAt_spray_projection {γ : ℝ → TangentBundle 𝓘(ℝ,E) M} {t : ℝ} |
| (hγ : IsMIntegralCurveAt γ (geodesicSpray (E := E)) t) : |
| HasMFDerivAt 𝓘(ℝ,ℝ) 𝓘(ℝ,E) (fun s => (γ s).1) t |
| ((1 : ℝ →L[ℝ] ℝ).smulRight (γ t).2) := by |
| have hd := (ContinuousLinearMap.fst ℝ E E).hasFDerivAt.comp_hasDerivAt t |
| (spray_chart_hasDerivAt hγ) |
| have ht := mem_extChartAt_source (I := 𝓘(ℝ,E)) (γ t).1 |
| have hself : (extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) (γ t) (γ t)).2 = |
| (γ t).2 := by |
| rw [tangent_chart_apply] |
| exact tangentCoordChange_self ht |
| change HasDerivAt (fun s => (extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) (γ t) (γ s)).1) |
| (extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) (γ t) (γ t)).2 t at hd |
| rw [hself] at hd |
| have hd' : HasDerivAt (extChartAt 𝓘(ℝ,E) (γ t).1 ∘ (fun s => (γ s).1)) |
| (γ t).2 t := by |
| simpa only [Function.comp_def,tangent_chart_apply] using hd |
| refine ⟨(FiberBundle.continuous_proj E (fun x : M => TangentSpace 𝓘(ℝ,E) x)).continuousAt.comp |
| hγ.continuousAt,?_⟩ |
| rw [writtenInExtChartAt,extChartAt_model_space_eq_id,ModelWithCorners.range_eq_univ] |
| change HasFDerivWithinAt (extChartAt 𝓘(ℝ,E) (γ t).1 ∘ (fun s => (γ s).1)) |
| ((1 : ℝ →L[ℝ] ℝ).smulRight (γ t).2) univ t |
| rw [hasFDerivWithinAt_univ] |
| convert! hd'.hasFDerivAt using 1 |
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| lemma spray_velocity_eq_mfderiv {γ : ℝ → TangentBundle 𝓘(ℝ,E) M} {t : ℝ} |
| (hγ : IsMIntegralCurveAt γ (geodesicSpray (E := E)) t) : |
| mfderiv 𝓘(ℝ,ℝ) 𝓘(ℝ,E) (fun s => (γ s).1) t (1 : ℝ) = (γ t).2 := by |
| have h := congrArg (fun f : TangentSpace 𝓘(ℝ,ℝ) t →L[ℝ] |
| TangentSpace 𝓘(ℝ,E) (γ t).1 => f (1 : ℝ)) (hasMFDerivAt_spray_projection hγ).mfderiv |
| change mfderiv 𝓘(ℝ,ℝ) 𝓘(ℝ,E) (fun s => (γ s).1) t (1 : ℝ) = |
| (1 : ℝ) • (γ t).2 at h |
| simpa only [one_smul] using h |
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| variable [T2Space M] [CompactSpace M] |
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| def sprayFlow (t : ℝ) (z : TangentBundle 𝓘(ℝ,E) M) : TangentBundle 𝓘(ℝ,E) M := |
| completeFlow exists_complete_spray_curve t z |
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| lemma sprayFlow_zero (z : TangentBundle 𝓘(ℝ,E) M) : sprayFlow 0 z = z := |
| completeFlow_zero exists_complete_spray_curve z |
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| lemma sprayFlow_curve (z : TangentBundle 𝓘(ℝ,E) M) : |
| IsMIntegralCurve (fun t => sprayFlow t z) (geodesicSpray (E := E)) := |
| completeFlow_curve exists_complete_spray_curve z |
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| lemma sprayFlow_add (s t : ℝ) (z : TangentBundle 𝓘(ℝ,E) M) : |
| sprayFlow (s+t) z = sprayFlow s (sprayFlow t z) := by |
| have : T2Space (TangentBundle 𝓘(ℝ,E) M) := bundle_totalSpace_t2 |
| exact completeFlow_add exists_complete_spray_curve contMDiff_geodesicSpray s t z |
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| lemma contMDiff_sprayFlow : |
| ContMDiff (𝓘(ℝ,ℝ).prod (𝓘(ℝ,E).prod 𝓘(ℝ,E))) (𝓘(ℝ,E).prod 𝓘(ℝ,E)) ∞ |
| (fun z : ℝ × TangentBundle 𝓘(ℝ,E) M => sprayFlow z.1 z.2) := by |
| have : T2Space (TangentBundle 𝓘(ℝ,E) M) := bundle_totalSpace_t2 |
| exact contMDiff_completeFlow exists_complete_spray_curve contMDiff_geodesicSpray |
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| lemma sprayFlow_speed (t : ℝ) (z : TangentBundle 𝓘(ℝ,E) M) : |
| ‖(sprayFlow t z).2‖ = ‖z.2‖ := by |
| have he := spray_speed_constant ((sprayFlow_curve z).isMIntegralCurveOn (Ioo (min t 0-1) (max t 0+1))) |
| (show t ∈ Ioo (min t 0-1) (max t 0+1) from |
| ⟨by linarith [min_le_left t 0],by linarith [le_max_left t 0]⟩) |
| (show (0:ℝ) ∈ Ioo (min t 0-1) (max t 0+1) from |
| ⟨by linarith [min_le_right t 0],by linarith [le_max_right t 0]⟩) |
| exact he.trans (congrArg (fun w : TangentBundle 𝓘(ℝ,E) M => ‖w.2‖) (sprayFlow_zero z)) |
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| end |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] |
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| omit [FiniteDimensional ℝ E] in |
| lemma tangent_chart_target_iff (a : TangentBundle 𝓘(ℝ,E) M) (q : E × E) : |
| q ∈ (extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) a).target ↔ |
| q.1 ∈ (extChartAt 𝓘(ℝ,E) a.1).target := by |
| rw [FiberBundle.extChartAt_target] |
| constructor |
| · exact fun h => h.1.1 |
| · intro h |
| refine ⟨⟨h,?_⟩,mem_univ _⟩ |
| simpa only [TangentBundle.trivializationAt_baseSet,← extChartAt_source 𝓘(ℝ,E),mem_preimage] using |
| (extChartAt 𝓘(ℝ,E) a.1).map_target h |
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| variable [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,E) x)] |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,E) ∞ E (fun x : M => TangentSpace 𝓘(ℝ,E) x)] |
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| lemma hasMFDerivAt_of_coordinate_spray (a : TangentBundle 𝓘(ℝ,E) M) |
| {q : ℝ → E × E} {t : ℝ} (hq : (q t).1 ∈ (extChartAt 𝓘(ℝ,E) a.1).target) |
| (hd : HasDerivAt q (coordinateGeodesicField (riemannianCoordinateMetric a.1) (q t)) t) : |
| HasMFDerivAt 𝓘(ℝ,ℝ) (𝓘(ℝ,E).prod 𝓘(ℝ,E)) |
| ((extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) a).symm ∘ q) t |
| ((1 : ℝ →L[ℝ] ℝ).smulRight |
| (geodesicSpray ((extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) a).symm (q t)))) := by |
| have hqt := (tangent_chart_target_iff a (q t)).mpr hq |
| apply hasMFDerivAt_of_chart_ode a hqt |
| rw [geodesicSpray_coordinates a _ ((tangent_chart_source_iff a _).mp |
| ((extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) a).map_target hqt)), |
| (extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) a).right_inv hqt] |
| exact hd |
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|
| omit [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,E) x)] |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,E) ∞ E (fun x : M => TangentSpace 𝓘(ℝ,E) x)] |
| [FiniteDimensional ℝ E] [IsManifold 𝓘(ℝ,E) ∞ M] in |
| lemma coordinate_geodesic_scale {g : E → E →L[ℝ] E →L[ℝ] ℝ} |
| {q : ℝ → E × E} {t c : ℝ} |
| (hq : HasDerivAt q (coordinateGeodesicField g (q (c*t))) (c*t)) : |
| HasDerivAt (fun s => ((q (c*s)).1,c • (q (c*s)).2)) |
| (coordinateGeodesicField g ((q (c*t)).1,c • (q (c*t)).2)) t := by |
| have hq₁ : HasDerivAt (fun s => (q s).1) (q (c*t)).2 (c*t) := |
| (ContinuousLinearMap.fst ℝ E E).hasFDerivAt.comp_hasDerivAt (c*t) hq |
| have hq₂ : HasDerivAt (fun s => (q s).2) |
| (-coordinateChristoffel g (q (c*t)).1 (q (c*t)).2 (q (c*t)).2) (c*t) := |
| (ContinuousLinearMap.snd ℝ E E).hasFDerivAt.comp_hasDerivAt (c*t) hq |
| have h₁ := hq₁.scomp t ((hasDerivAt_id t).const_mul c) |
| have h₂ := (hq₂.scomp t ((hasDerivAt_id t).const_mul c)).const_smul c |
| have h := h₁.prodMk h₂ |
| convert! h using 1 |
| simp only [coordinateGeodesicField,coordinateChristoffel_smul_left, |
| coordinateChristoffel_smul_right,mul_one,smul_neg] |
|
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| variable [T2Space M] [CompactSpace M] |
|
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| lemma sprayFlow_eq_coordinate_curve (a : TangentBundle 𝓘(ℝ,E) M) |
| {q : ℝ → E × E} {r : ℝ} (hr : 0 < r) |
| (himg : ∀ t ∈ Ioo (-r) r, (q t).1 ∈ (extChartAt 𝓘(ℝ,E) a.1).target) |
| (hode : ∀ t ∈ Ioo (-r) r, |
| HasDerivAt q (coordinateGeodesicField (riemannianCoordinateMetric a.1) (q t)) t) |
| {z : TangentBundle 𝓘(ℝ,E) M} |
| (hq0 : (extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) a).symm (q 0) = z) |
| {t : ℝ} (ht : t ∈ Ioo (-r) r) : |
| sprayFlow t z = (extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) a).symm (q t) := by |
| have : T2Space (TangentBundle 𝓘(ℝ,E) M) := bundle_totalSpace_t2 |
| apply isMIntegralCurveOn_Ioo_eqOn_of_contMDiff_boundaryless |
| (t₀ := 0) ⟨neg_lt_zero.mpr hr,hr⟩ (contMDiff_geodesicSpray.of_le (by simp)) |
| ((sprayFlow_curve z).isMIntegralCurveOn _) _ _ ht |
| · intro s hs |
| exact (hasMFDerivAt_of_coordinate_spray a (himg s hs) (hode s hs)).hasMFDerivWithinAt |
| · exact (sprayFlow_zero z).trans hq0.symm |
|
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| end WeakMTWTransport |
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| end |
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| end OAI |
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