| import Mathlib |
| import OAI.Analysis.BiholderTransport.Coordinates.CoordinateMetricNorm |
| import OAI.Analysis.BiholderTransport.Volume.RiemannianCalibration |
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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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| namespace WeakMTWTransport |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] |
| {M : Type*} [MetricSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,E) x)] |
| [IsRiemannianManifold 𝓘(ℝ,E) M] |
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| omit [IsRiemannianManifold 𝓘(ℝ,E) M] in |
| lemma coordinate_flow_path_length |
| {g : E → E →L[ℝ] E →L[ℝ] ℝ} {S : Set E} (hS : IsOpen S) |
| (hg : ContDiffOn ℝ ∞ g S) (hpos : ∀ x ∈ S, ∀ v : E, v ≠ 0 → 0 < g x v v) |
| (hsym : ∀ x ∈ S, ∀ u v, g x u v = g x v u) |
| {c : M} (hSto : S ⊆ (extChartAt 𝓘(ℝ,E) c).target) |
| (hmetric : ∀ z ∈ S, ∀ u v : E, g z u v = |
| inner ℝ ((trivializationAt E (fun x : M => TangentSpace 𝓘(ℝ,E) x) c).symmL ℝ |
| ((extChartAt 𝓘(ℝ,E) c).symm z) u) |
| ((trivializationAt E (fun x : M => TangentSpace 𝓘(ℝ,E) x) c).symmL ℝ |
| ((extChartAt 𝓘(ℝ,E) c).symm z) v)) |
| {η V : ℝ → E} {r : ℝ} (hη : ContDiffOn ℝ ∞ η (Ioo (-r) r)) |
| (himg : MapsTo η (Ioo (-r) r) S) |
| (hode : ∀ t ∈ Ioo (-r) r, HasDerivAt η (V t) t ∧ |
| HasDerivAt V (-coordinateChristoffel g (η t) (V t) (V t)) t) |
| {τ : ℝ} (hτ : 0 ≤ τ) (hτr : τ < r) : |
| pathELength 𝓘(ℝ,E) ((extChartAt 𝓘(ℝ,E) c).symm ∘ η) 0 τ = |
| ENNReal.ofReal (τ * Real.sqrt (g (η 0) (V 0) (V 0))) := by |
| have hr : 0 < r := hτ.trans_lt hτr |
| have h0 : (0:ℝ) ∈ Ioo (-r) r := ⟨by linarith,hr⟩ |
| have hsub : Icc 0 τ ⊆ Ioo (-r) r := fun t ht => ⟨by linarith [ht.1],ht.2.trans_lt hτr⟩ |
| rw [inverse_coordinate_path_length ((hη.mono hsub).of_le (by simp)) |
| (fun t ht => hSto (himg (hsub ht))) g (fun t ht => hmetric _ (himg (hsub ht)))] |
| have hi : ∀ y ∈ S, (g y).IsInvertible := fun y hy => metricDual_isInvertible (g y) (hpos y hy) |
| have hc : ∀ t ∈ Icc 0 τ, g (η t) (deriv η t) (deriv η t) = g (η 0) (V 0) (V 0) := by |
| intro t ht |
| rw [(hode t (hsub ht)).1.deriv] |
| exact coordinate_geodesic_speed_constant |
| (fun s hs => (hg.contDiffAt (hS.mem_nhds (himg hs))).differentiableAt (by simp)) |
| (fun s hs => hi _ (himg hs)) (fun s hs => hsym _ (himg hs)) hode (hsub ht) h0 |
| calc |
| _ = ∫⁻ t in Icc (0:ℝ) τ, ENNReal.ofReal (Real.sqrt (g (η 0) (V 0) (V 0))) := by |
| apply setLIntegral_congr_fun measurableSet_Icc |
| intro t ht |
| dsimp only |
| rw [hc t ht] |
| _ = _ := by rw [setLIntegral_const,Real.volume_Icc,sub_zero,ENNReal.ofReal_mul hτ,mul_comm] |
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| lemma riemannian_normal_chart_dist_lower |
| {g : E → E →L[ℝ] E →L[ℝ] ℝ} {S : Set E} (hS : IsOpen S) |
| (hg : ContDiffOn ℝ ∞ g S) (hpos : ∀ x ∈ S, ∀ v : E, v ≠ 0 → 0 < g x v v) |
| (hsym : ∀ x ∈ S, ∀ u v, g x u v = g x v u) |
| {c : M} |
| (hmetric : ∀ z ∈ S, ∀ u v : E, g z u v = |
| inner ℝ ((trivializationAt E (fun x : M => TangentSpace 𝓘(ℝ,E) x) c).symmL ℝ |
| ((extChartAt 𝓘(ℝ,E) c).symm z) u) |
| ((trivializationAt E (fun x : M => TangentSpace 𝓘(ℝ,E) x) c).symmL ℝ |
| ((extChartAt 𝓘(ℝ,E) c).symm z) v)) |
| {Ψ W : ℝ × E → E} {r : ℝ} |
| (hΨ : ContDiffOn ℝ ∞ Ψ (Ioo (-r) r ×ˢ Metric.ball 0 r)) |
| (hW : ContDiffOn ℝ ∞ W (Ioo (-r) r ×ˢ Metric.ball 0 r)) |
| (himg : ∀ t ∈ Ioo (-r) r, ∀ v ∈ Metric.ball 0 r, Ψ (t,v) ∈ S) |
| (hode : ∀ t ∈ Ioo (-r) r, ∀ v ∈ Metric.ball 0 r, |
| HasDerivAt (fun q => Ψ (q,v)) (W (t,v)) t ∧ |
| HasDerivAt (fun q => W (q,v)) |
| (-coordinateChristoffel g (Ψ (t,v)) (W (t,v)) (W (t,v))) t) |
| {x : E} (h0 : 0 ∈ Ioo (-r) r) |
| (hΨ0 : ∀ v ∈ Metric.ball 0 r, Ψ (0,v) = x) |
| (hW0 : ∀ v ∈ Metric.ball 0 r, W (0,v) = v) |
| {τ : ℝ} (hτ : τ ∈ Ioo (-r) r) (hτnonneg : 0 ≤ τ) |
| (e : OpenPartialHomeomorph E E) (hef : (e : E → E) = fun v => Ψ (τ,v)) |
| (hes : 0 ∈ e.source) (hesub : e.source ⊆ Metric.ball 0 r) |
| (hed : ContDiffOn ℝ ∞ e e.source) (hei : ContDiffOn ℝ ∞ e.symm e.target) |
| {a y : M} (he0 : e 0 = extChartAt 𝓘(ℝ,E) c a) |
| {δ : ℝ} (hball : Metric.ball a δ ⊆ |
| (extChartAt 𝓘(ℝ,E) c).source ∩ (extChartAt 𝓘(ℝ,E) c) ⁻¹' e.target) |
| (hdist : dist a y < δ) : |
| τ * Real.sqrt (g x (e.symm (extChartAt 𝓘(ℝ,E) c y)) |
| (e.symm (extChartAt 𝓘(ℝ,E) c y))) ≤ dist a y := by |
| apply riemannian_local_dist_lower_bound (I := 𝓘(ℝ,E)) hball hdist |
| intro γ hγ hγ0 hγ1 hγimg |
| let η : ℝ → E := (extChartAt 𝓘(ℝ,E) c) ∘ γ |
| have hη : ContDiffOn ℝ 1 η (Icc 0 1) := |
| contMDiffOn_iff_contDiffOn.mp |
| ((contMDiffOn_extChartAt (I := 𝓘(ℝ,E)) (n := 1) (x := c)).comp hγ (fun t ht => by |
| change γ t ∈ (chartAt E c).source |
| simpa only [extChartAt_source] using (hγimg ht).1)) |
| have hηimg : MapsTo η (Icc 0 1) e.target := fun t ht => (hγimg ht).2 |
| have hηS : ∀ t ∈ Icc 0 1, η t ∈ S := by |
| intro t ht |
| have hh := himg τ hτ (e.symm (η t)) (hesub (e.map_target (hηimg ht))) |
| have hval := congrFun hef (e.symm (η t)) |
| rw [←hval,e.right_inv (hηimg ht)] at hh |
| exact hh |
| have hb := coordinate_normal_chart_radial_bound hS hg hpos hsym hΨ hW himg hode |
| h0 hΨ0 hW0 hτ hτnonneg e hef hes hesub hed hei (by norm_num : (0:ℝ) ≤ 1) |
| hη hηimg (by simp only [η,Function.comp_apply,hγ0,he0]) |
| have hlen := coordinate_path_length hγ (fun t ht => (hγimg ht).1) g (by |
| intro t ht u v |
| have hh := hmetric (η t) (hηS t ht) u v |
| have hinner := congrArg (fun z : M => |
| inner ℝ ((trivializationAt E (fun x : M => TangentSpace 𝓘(ℝ,E) x) c).symmL ℝ z u) |
| ((trivializationAt E (fun x : M => TangentSpace 𝓘(ℝ,E) x) c).symmL ℝ z v)) |
| ((extChartAt 𝓘(ℝ,E) c).left_inv (hγimg ht).1) |
| exact hh.trans hinner) |
| rw [hlen] |
| simpa only [η,Function.comp_apply,hγ1] using hb |
| |
| lemma riemannian_coordinate_normal_chart_dist |
| {g : E → E →L[ℝ] E →L[ℝ] ℝ} {S : Set E} (hS : IsOpen S) |
| (hg : ContDiffOn ℝ ∞ g S) (hpos : ∀ x ∈ S, ∀ v : E, v ≠ 0 → 0 < g x v v) |
| (hsym : ∀ x ∈ S, ∀ u v, g x u v = g x v u) |
| {c : M} (hSto : S ⊆ (extChartAt 𝓘(ℝ,E) c).target) |
| (hmetric : ∀ z ∈ S, ∀ u v : E, g z u v = |
| inner ℝ ((trivializationAt E (fun x : M => TangentSpace 𝓘(ℝ,E) x) c).symmL ℝ |
| ((extChartAt 𝓘(ℝ,E) c).symm z) u) |
| ((trivializationAt E (fun x : M => TangentSpace 𝓘(ℝ,E) x) c).symmL ℝ |
| ((extChartAt 𝓘(ℝ,E) c).symm z) v)) |
| {Ψ W : ℝ × E → E} {r : ℝ} |
| (hΨ : ContDiffOn ℝ ∞ Ψ (Ioo (-r) r ×ˢ Metric.ball 0 r)) |
| (hW : ContDiffOn ℝ ∞ W (Ioo (-r) r ×ˢ Metric.ball 0 r)) |
| (himg : ∀ t ∈ Ioo (-r) r, ∀ v ∈ Metric.ball 0 r, Ψ (t,v) ∈ S) |
| (hode : ∀ t ∈ Ioo (-r) r, ∀ v ∈ Metric.ball 0 r, |
| HasDerivAt (fun q => Ψ (q,v)) (W (t,v)) t ∧ |
| HasDerivAt (fun q => W (q,v)) |
| (-coordinateChristoffel g (Ψ (t,v)) (W (t,v)) (W (t,v))) t) |
| {a y : M} (h0 : 0 ∈ Ioo (-r) r) |
| (hΨ0 : ∀ v ∈ Metric.ball 0 r, Ψ (0,v) = extChartAt 𝓘(ℝ,E) c a) |
| (hW0 : ∀ v ∈ Metric.ball 0 r, W (0,v) = v) |
| {τ : ℝ} (hτ : τ ∈ Ioo (-r) r) (hτnonneg : 0 ≤ τ) |
| (e : OpenPartialHomeomorph E E) (hef : (e : E → E) = fun v => Ψ (τ,v)) |
| (he0 : e 0 = extChartAt 𝓘(ℝ,E) c a) |
| (hes : 0 ∈ e.source) (hesub : e.source ⊆ Metric.ball 0 r) |
| (hed : ContDiffOn ℝ ∞ e e.source) (hei : ContDiffOn ℝ ∞ e.symm e.target) |
| {δ : ℝ} (hball : Metric.ball a δ ⊆ |
| (extChartAt 𝓘(ℝ,E) c).source ∩ (extChartAt 𝓘(ℝ,E) c) ⁻¹' e.target) |
| (hdist : dist a y < δ) : |
| dist a y = τ * Real.sqrt (g (extChartAt 𝓘(ℝ,E) c a) |
| (e.symm (extChartAt 𝓘(ℝ,E) c y)) (e.symm (extChartAt 𝓘(ℝ,E) c y))) := by |
| apply le_antisymm _ (riemannian_normal_chart_dist_lower hS hg hpos hsym hmetric |
| hΨ hW himg hode h0 hΨ0 hW0 hτ hτnonneg e hef hes hesub hed hei he0 hball hdist) |
| have hδ : 0 < δ := dist_nonneg.trans_lt hdist |
| have ha := hball (Metric.mem_ball_self hδ) |
| have hy := hball (show y ∈ Metric.ball a δ by simpa only [Metric.mem_ball,dist_comm] using hdist) |
| let v := e.symm (extChartAt 𝓘(ℝ,E) c y) |
| have hv : v ∈ Metric.ball 0 r := hesub (e.map_target hy.2) |
| let η : ℝ → E := fun t => Ψ (t,v) |
| have hη : ContDiffOn ℝ ∞ η (Ioo (-r) r) := |
| hΨ.comp (contDiffOn_id.prodMk contDiffOn_const) (fun t ht => ⟨ht,hv⟩) |
| have hsub : Icc 0 τ ⊆ Ioo (-r) r := fun t ht => ⟨h0.1.trans_le ht.1,ht.2.trans_lt hτ.2⟩ |
| have hγ : ContMDiffOn 𝓘(ℝ,ℝ) 𝓘(ℝ,E) 1 |
| ((extChartAt 𝓘(ℝ,E) c).symm ∘ η) (Icc 0 τ) := |
| (contMDiffOn_extChartAt_symm (n := 1) c).comp |
| (contMDiffOn_iff_contDiffOn.mpr ((hη.mono hsub).of_le (by simp))) |
| (fun t ht => hSto (himg t (hsub ht) v hv)) |
| have hγ0 : ((extChartAt 𝓘(ℝ,E) c).symm ∘ η) 0 = a := by |
| simp only [Function.comp_apply,η,hΨ0 v hv,(extChartAt 𝓘(ℝ,E) c).left_inv ha.1] |
| have hγτ : ((extChartAt 𝓘(ℝ,E) c).symm ∘ η) τ = y := by |
| have heq := congrFun hef v |
| simp only [Function.comp_apply,η,←heq,v,e.right_inv hy.2, |
| (extChartAt 𝓘(ℝ,E) c).left_inv hy.1] |
| have hh := riemannianEDist_le_pathELength (I := 𝓘(ℝ,E)) hγ hγ0 hγτ hτnonneg |
| rw [←IsRiemannianManifold.out,edist_dist] at hh |
| have hlen := coordinate_flow_path_length hS hg hpos hsym hSto hmetric hη |
| (fun t ht => himg t ht v hv) (fun t ht => hode t ht v hv) hτnonneg hτ.2 |
| rw [hlen] at hh |
| have hc : 0 ≤ τ * Real.sqrt (g (extChartAt 𝓘(ℝ,E) c a) v v) := mul_nonneg hτnonneg (Real.sqrt_nonneg _) |
| have hh' : ENNReal.ofReal (dist a y) ≤ |
| ENNReal.ofReal (τ * Real.sqrt (g (extChartAt 𝓘(ℝ,E) c a) v v)) := by |
| simpa only [η,hΨ0 v hv,hW0 v hv] using hh |
| exact (ENNReal.ofReal_le_ofReal_iff hc).mp hh' |
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| variable [IsContMDiffRiemannianBundle 𝓘(ℝ,E) ∞ E |
| (fun x : M => TangentSpace 𝓘(ℝ,E) x)] |
| end WeakMTWTransport |
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| end |
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| end OAI |
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