| import Mathlib |
| import OAI.Analysis.BiholderTransport.Geodesics.SprayProjection |
| import OAI.Analysis.BiholderTransport.Coordinates.JointNormalChart |
| import OAI.Analysis.BiholderTransport.Coordinates.NormalGradient |
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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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| 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] |
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|
| lemma exists_local_metric_gradient (a : M) : |
| ∃ δ τ : ℝ, 0 < δ ∧ 0 < τ ∧ |
| Metric.ball a δ ⊆ (extChartAt 𝓘(ℝ,E) a).source ∧ |
| ∀ b ∈ Metric.ball a δ, ∀ y ∈ Metric.ball a δ, |
| ∃ z : TangentBundle 𝓘(ℝ,E) M, |
| z.1 = b ∧ (sprayFlow τ z).1 = y ∧ dist b y = τ * ‖z.2‖ ∧ |
| HasFDerivAt (fun w : E => dist b ((extChartAt 𝓘(ℝ,E) a).symm w)^2 / 2) |
| (τ • riemannianCoordinateMetric a (extChartAt 𝓘(ℝ,E) a y) |
| ((extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) |
| (⟨a,0⟩ : TangentBundle 𝓘(ℝ,E) M) (sprayFlow τ z)).2)) |
| (extChartAt 𝓘(ℝ,E) a y) := by |
| obtain ⟨δ,r,τ,hδ,hr,hτ,hτr,Φ,e,hΦ,hΦ0,hode,he,hesub,hed,hei,hsrc,hdiag,htarget,hdist⟩ := |
| exists_uniform_metric_normal_coordinates (E := E) a |
| refine ⟨δ,τ,hδ,hτ,hsrc,?_⟩ |
| intro b hb y hy |
| let c := extChartAt 𝓘(ℝ,E) a |
| let A : TangentBundle 𝓘(ℝ,E) M := ⟨a,0⟩ |
| let χ := extChartAt (𝓘(ℝ,E).prod 𝓘(ℝ,E)) A |
| let g := riemannianCoordinateMetric (E := E) a |
| have hp : ∀ z : E × E, (e z).1 = z.1 := by intro z; rw [he] |
| let f := fiberwiseHomeomorph e hp (c b) |
| let v := f.symm (c y) |
| have hyt : c y ∈ f.target := htarget b hb y hy |
| have hvs : v ∈ f.source := f.map_target hyt |
| have hcb : dist (c b) (c a) < r := |
| (max_lt_iff.mp (show max (dist (c b) (c a)) (dist (0:E) 0) < r from hesub (hdiag b hb))).1 |
| have hball : ∀ w ∈ Metric.ball (0:E) r, (c b,w) ∈ Metric.ball (c a,0) r := by |
| intro w hw |
| exact max_lt hcb hw |
| have hv : v ∈ Metric.ball (0:E) r := |
| (max_lt_iff.mp (show max (dist (c b) (c a)) (dist v 0) < r from hesub hvs)).2 |
| let Ψ : ℝ × E → E := fun z => (Φ (z.1,(c b,z.2))).1 |
| let W : ℝ × E → E := fun z => (Φ (z.1,(c b,z.2))).2 |
| have hΦc := hΦ.comp (contDiffOn_fst.prodMk (contDiffOn_const.prodMk contDiffOn_snd)) |
| (show MapsTo (fun z : ℝ × E => (z.1,(c b,z.2))) |
| (Ioo (-r) r ×ˢ Metric.ball 0 r) (Ioo (-r) r ×ˢ Metric.ball (c a,0) r) from |
| fun z hz => ⟨hz.1,hball z.2 hz.2⟩) |
| have hf : (f : E → E) = fun w => Ψ (τ,w) := by |
| funext w |
| change (e (c b,w)).2 = _ |
| rw [he] |
| have ht : τ ∈ Ioo (-r) r := ⟨by linarith,hτr⟩ |
| have h0 : (0:ℝ) ∈ Ioo (-r) r := ⟨by linarith,hr⟩ |
| have hΨv : Ψ (τ,v) = c y := (congrFun hf v).symm.trans (f.right_inv hyt) |
| have hG : ∀ w : E, g (c y) (W (τ,v)) (fderiv ℝ f v w) = τ*g (c b) v w := by |
| intro w |
| rw [hf,←hΨv] |
| exact coordinate_endpoint_gauss (Ψ := Ψ) (W := W) (x := c b) (g := g) |
| (isOpen_extChartAt_target a) |
| (contDiffOn_riemannianCoordinateMetric a) |
| (fun _ hx => riemannianCoordinateMetric_isInvertible hx) |
| (fun x _ u v => riemannianCoordinateMetric_symm a x u v) |
| (show ContDiffOn ℝ ∞ Ψ (Ioo (-r) r ×ˢ Metric.ball 0 r) from hΦc.fst) |
| (show ContDiffOn ℝ ∞ W (Ioo (-r) r ×ˢ Metric.ball 0 r) from hΦc.snd) |
| (fun t ht w hw => (hode t ht (c b,w) (hball w hw)).1) |
| (fun t ht w hw => ⟨((hode t ht (c b,w) (hball w hw)).2).fst, |
| ((hode t ht (c b,w) (hball w hw)).2).snd⟩) h0 |
| (fun w hw => congrArg Prod.fst (hΦ0 (c b,w) (hball w hw))) |
| (fun w hw => congrArg Prod.snd (hΦ0 (c b,w) (hball w hw))) ht hv w |
| have hfd := ((fiberwiseHomeomorph_contDiffOn hp (c b) hed).contDiffAt |
| (f.open_source.mem_nhds hvs)).differentiableAt (by simp) |
| have hfi := ((fiberwiseHomeomorph_symm_contDiffOn hp (c b) hei).contDiffAt |
| (f.open_target.mem_nhds hyt)).differentiableAt (by simp) |
| have hD := normal_action_hasFDerivAt f hyt hfd hfi (g (c b)) (g (c y)) |
| (riemannianCoordinateMetric_symm a (c b)) (W (τ,v)) τ hG |
| have hnear : ∀ᶠ w : E in 𝓝 (c y), w ∈ c.target ∧ c.symm w ∈ Metric.ball a δ := by |
| have hcy : c y ∈ c.target := c.map_source (hsrc hy) |
| have hynear := ((continuousOn_extChartAt_symm a).continuousAt ((isOpen_extChartAt_target a).mem_nhds hcy)).preimage_mem_nhds |
| (by rw [c.left_inv (hsrc hy)]; exact Metric.isOpen_ball.mem_nhds hy) |
| filter_upwards [(isOpen_extChartAt_target a).mem_nhds hcy, hynear] with w hw hwy |
| exact ⟨hw,hwy⟩ |
| have hdistEq : (fun w : E => dist b (c.symm w)^2/2) =ᶠ[𝓝 (c y)] |
| (fun w => (τ^2/2)*g (c b) (f.symm w) (f.symm w)) := by |
| filter_upwards [hnear] with w hw |
| have hwt : w ∈ f.target := by |
| change (c b,w) ∈ e.target |
| have hh : (c b,c (c.symm w)) ∈ e.target := htarget b hb (c.symm w) hw.2 |
| simpa only [c.right_inv hw.1] using hh |
| have hh := hdist b hb (c.symm w) hw.2 (f.symm w) (f.map_target hwt) |
| (show (Φ (τ,(c b,f.symm w))).1 = c (c.symm w) by |
| rw [c.right_inv hw.1] |
| exact (congrFun hf _).symm.trans (f.right_inv hwt)) |
| have hn : 0 ≤ g (c b) (f.symm w) (f.symm w) := by |
| by_cases hh : f.symm w = 0 |
| · simp only [hh,map_zero,le_refl] |
| · exact (riemannianCoordinateMetric_positive (c.map_source (hsrc hb)) hh).le |
| rw [hh,mul_pow,Real.sq_sqrt hn] |
| ring |
| let z := χ.symm (c b,v) |
| have hzt : (c b,v) ∈ χ.target := (tangent_chart_target_iff A _).mpr (c.map_source (hsrc hb)) |
| have hzchart : χ z = (c b,v) := χ.right_inv hzt |
| have hzsrc : z ∈ χ.source := χ.map_target hzt |
| have hzb : z.1 = b := by |
| have hh := congrArg Prod.fst hzchart |
| rw [tangent_chart_apply] at hh |
| exact c.injOn ((tangent_chart_source_iff A z).mp hzsrc) (hsrc hb) hh |
| have hflow := sprayFlow_eq_coordinate_curve A hr |
| (fun t ht' => (hode t ht' (c b,v) (hball v hv)).1) |
| (fun t ht' => (hode t ht' (c b,v) (hball v hv)).2) |
| (show χ.symm (Φ (0,(c b,v))) = z by rw [hΦ0 (c b,v) (hball v hv)]) ht |
| have hflowchart : χ (sprayFlow τ z) = Φ (τ,(c b,v)) := by |
| rw [hflow,χ.right_inv ((tangent_chart_target_iff A _).mpr (hode τ ht (c b,v) (hball v hv)).1)] |
| have hflowsrc : sprayFlow τ z ∈ χ.source := by |
| rw [hflow] |
| exact χ.map_target ((tangent_chart_target_iff A _).mpr (hode τ ht (c b,v) (hball v hv)).1) |
| have hfy : (sprayFlow τ z).1 = y := by |
| apply c.injOn ((tangent_chart_source_iff A _).mp hflowsrc) (hsrc hy) |
| exact (congrArg Prod.fst (tangent_chart_apply A (sprayFlow τ z))).symm.trans |
| ((congrArg Prod.fst hflowchart).trans hΨv) |
| refine ⟨z,hzb,hfy,?_,?_⟩ |
| · have hh := hdist b hb y hy v hvs hΨv |
| have hn := coordinate_spray_energy A z ((tangent_chart_source_iff A z).mp hzsrc) |
| rw [hzchart] at hn |
| rw [hh,hn,Real.sqrt_sq (norm_nonneg _)] |
| · have hh := hD.congr_of_eventuallyEq hdistEq |
| rw [show (χ (sprayFlow τ z)).2 = W (τ,v) from congrArg Prod.snd hflowchart] |
| exact hh |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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