| import Mathlib |
| import OAI.Analysis.BiholderTransport.Geodesics.GeodesicBasics |
| import OAI.Analysis.BiholderTransport.Regularity.RiemannianMetricSegment |
| import OAI.Analysis.BiholderTransport.Geodesics.NoCorner |
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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 |
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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 sprayFlow_minimizing_subinterval (z : TangentBundle 𝓘(ℝ,E) M) {s t : ℝ} |
| (hs : 0 ≤ s) (hst : s ≤ t) |
| (hmin : dist z.1 (sprayFlow t z).1 = t * ‖z.2‖) : |
| dist (sprayFlow s z).1 (sprayFlow t z).1 = (t-s) * ‖z.2‖ := by |
| have hle := dist_sprayFlow_le z s t |
| rw [abs_of_nonpos (sub_nonpos.mpr hst),neg_sub] at hle |
| apply le_antisymm hle |
| have hlow := dist_triangle z.1 (sprayFlow s z).1 (sprayFlow t z).1 |
| rw [hmin,sprayFlow_minimizing_prefix z hs hst hmin] at hlow |
| nlinarith |
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| end |
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| 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] |
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| lemma exists_local_minimizingVectors (x : M) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ y ∈ Metric.ball x δ, |
| ∃ p ∈ minimizingVectors (n := n) x, riemannianExp x p = y := by |
| obtain ⟨δ,hδ,H⟩ := exists_local_cost_gradient (E := Model n) x |
| refine ⟨δ,hδ,?_⟩ |
| intro y hy |
| obtain ⟨q,hqx,hqn,hqb,_⟩ := H y hy x (Metric.mem_ball_self hδ) |
| rcases q with ⟨a,u⟩ |
| change a=x at hqx |
| subst a |
| have he : riemannianExp x (-u) = y := by |
| rw [show -u = (-1:ℝ) • u by simp,riemannianExp_smul] |
| exact hqb |
| refine ⟨-u,?_,he⟩ |
| change dist x (riemannianExp x (-u)) = ‖-u‖ |
| rw [he,norm_neg,hqn,dist_comm] |
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| lemma exists_minimizing_vector (x y : M) : |
| ∃ p ∈ minimizingVectors (n := n) x, riemannianExp x p = y := by |
| classical |
| by_cases hxy : x=y |
| · subst y |
| exact ⟨0,by simp [minimizingVectors],riemannianExp_zero x⟩ |
| have hD : 0 < dist x y := dist_pos.mpr hxy |
| obtain ⟨γ,hγ,hγ0,hγD⟩ := riemannian_exists_metric_segment (I := 𝓘(ℝ,Model n)) x y |
| have hd (s : Icc (0:ℝ) (dist x y)) : dist x (γ s) = s.1 := by |
| have hh := hγ.dist_eq (⟨0,⟨le_rfl,dist_nonneg⟩⟩ : Icc (0:ℝ) (dist x y)) s |
| simpa only [hγ0,Subtype.dist_eq,Real.dist_eq,zero_sub,abs_neg,abs_of_nonneg s.2.1] using hh |
| let R : Set M := riemannianExp (n := n) x '' minimizingVectors x |
| have hR : IsCompact R := (isCompact_minimizingVectors x).image (continuous_riemannianExp x) |
| let U : Set (Icc (0:ℝ) (dist x y)) := γ ⁻¹' R |
| have hU : IsCompact U := (hR.isClosed.preimage hγ.continuous).isCompact |
| obtain ⟨δ,hδ,Hlocal⟩ := exists_local_minimizingVectors (n := n) x |
| let a : ℝ := min (dist x y/2) (δ/2) |
| have ha : 0 < a := lt_min (by positivity) (by positivity) |
| have haD : a ≤ dist x y := (min_le_left _ _).trans (by linarith) |
| have haδ : a < δ := (min_le_right _ _).trans_lt (by linarith) |
| let s₀ : Icc (0:ℝ) (dist x y) := ⟨a,ha.le,haD⟩ |
| have hs₀ : s₀ ∈ U := by |
| obtain ⟨p,hp,he⟩ := Hlocal (γ s₀) (by rw [Metric.mem_ball,dist_comm,hd]; exact haδ) |
| exact ⟨p,hp,he⟩ |
| obtain ⟨s,hs,hsmax⟩ := hU.exists_isMaxOn ⟨s₀,hs₀⟩ continuous_subtype_val.continuousOn |
| have hs0 : 0 < s.1 := ha.trans_le (hsmax hs₀) |
| have hsD : s.1 = dist x y := by |
| by_contra hsne |
| have hslt : s.1 < dist x y := lt_of_le_of_ne s.2.2 hsne |
| obtain ⟨p,hp,hps⟩ := hs |
| have hpn : ‖p‖ = s.1 := by |
| change dist x (riemannianExp x p) = ‖p‖ at hp |
| rw [hps,hd] at hp |
| exact hp.symm |
| let z : TangentBundle 𝓘(ℝ,Model n) M := ⟨x,s.1⁻¹ • p⟩ |
| have hzn : ‖z.2‖ = 1 := by |
| change ‖s.1⁻¹ • p‖ = 1 |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hs0),hpn,inv_mul_cancel₀ hs0.ne'] |
| have hzflow : (sprayFlow s.1 z).1 = γ s := by |
| rw [←riemannianExp_smul] |
| rw [smul_smul,mul_inv_cancel₀ hs0.ne',one_smul,hps] |
| have hzmin : dist z.1 (sprayFlow s.1 z).1 = s.1 * ‖z.2‖ := by |
| rw [hzflow,hzn,mul_one]; exact hd s |
| obtain ⟨η,hη,Hext⟩ := exists_spray_local_extension (E := Model n) (γ s) |
| let ε : ℝ := min (η/2) (min (s.1/2) ((dist x y-s.1)/2)) |
| have hε : 0 < ε := lt_min (by positivity) (lt_min (by positivity) (by linarith)) |
| have hεη : ε < η := (min_le_left _ _).trans_lt (by linarith) |
| have hεs : ε < s.1 := ((min_le_right _ _).trans (min_le_left _ _)).trans_lt (by linarith) |
| have hεD : s.1+ε < dist x y := by |
| have h := (min_le_right (η/2) (min (s.1/2) ((dist x y-s.1)/2))).trans (min_le_right _ _) |
| change ε ≤ (dist x y-s.1)/2 at h |
| linarith |
| let s' : Icc (0:ℝ) (dist x y) := ⟨s.1+ε,by linarith,hεD.le⟩ |
| let q := sprayFlow s.1 z |
| have hqn : ‖q.2‖ = 1 := by rw [sprayFlow_speed]; exact hzn |
| have hqback : sprayFlow (-ε) q = sprayFlow (s.1-ε) z := by |
| rw [←sprayFlow_add]; congr 1; ring |
| have hpre : dist (sprayFlow (-ε) q).1 q.1 = ε := by |
| rw [hqback] |
| have h := sprayFlow_minimizing_subinterval z (s := s.1-ε) (by linarith) (by linarith) hzmin |
| rw [hzn,mul_one] at h |
| simpa only [sub_sub_cancel] using h |
| have hnext : dist q.1 (γ s') = ε := by |
| change dist (sprayFlow s.1 z).1 _ = _ |
| rw [hzflow,hγ.dist_eq,Subtype.dist_eq,Real.dist_eq] |
| change |s.1-(s.1+ε)| = ε |
| rw [show s.1-(s.1+ε) = -ε by ring,abs_neg,abs_of_pos hε] |
| have hlong : dist (sprayFlow (-ε) q).1 (γ s') = 2*ε := by |
| apply le_antisymm |
| · exact (dist_triangle _ q.1 _).trans (by rw [hpre,hnext]; linarith) |
| · have h := dist_triangle x (sprayFlow (-ε) q).1 (γ s') |
| rw [hd,hqback] at h |
| have hpref := sprayFlow_minimizing_prefix z (s := s.1-ε) (by linarith) (by linarith) hzmin |
| rw [hzn,mul_one] at hpref |
| change dist x (sprayFlow (s.1-ε) z).1 = s.1-ε at hpref |
| rw [hpref] at h |
| change s.1+ε ≤ _ at h |
| rw [hqback] |
| linarith |
| have hext := Hext q hzflow hqn ε hε hεη (γ s') hpre hnext hlong |
| have hznext : (sprayFlow (s.1+ε) z).1 = γ s' := by |
| rw [add_comm,sprayFlow_add]; exact hext |
| have hs' : s' ∈ U := by |
| refine ⟨(s.1+ε) • z.2,?_,?_⟩ |
| · change dist x (riemannianExp x ((s.1+ε) • z.2)) = ‖(s.1+ε) • z.2‖ |
| rw [riemannianExp_smul,hznext,hd,norm_smul,Real.norm_eq_abs,abs_of_pos (by linarith : 0 < s.1+ε),hzn,mul_one] |
| · rw [riemannianExp_smul]; exact hznext |
| have hh := hsmax hs' |
| change s.1+ε ≤ s.1 at hh |
| linarith |
| obtain ⟨p,hp,he⟩ := hs |
| refine ⟨p,hp,he.trans ?_⟩ |
| have hsEq : s = ⟨dist x y,⟨dist_nonneg,le_rfl⟩⟩ := Subtype.ext hsD |
| rw [hsEq,hγD] |
|
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| end WeakMTWTransport |
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| end |
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| end OAI |
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