| import Mathlib |
| import OAI.Analysis.BiholderTransport.Geodesics.SprayMinimizerUnique |
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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 {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 riemannianExp_interior_unique {x : M} {p q : TangentSpace 𝓘(ℝ,Model n) x} |
| (hp : p ∈ injectivityDomain x) (hq : q ∈ minimizingVectors x) |
| (he : riemannianExp x p = riemannianExp x q) : p=q := by |
| have hpmin := injectivityDomain_subset_minimizingVectors x hp |
| have hn : ‖p‖=‖q‖ := hpmin.symm.trans ((congrArg (dist x) he).trans hq) |
| by_cases hp0 : p=0 |
| · have hq0 : q=0 := norm_eq_zero.mp (by rw [←hn,hp0,norm_zero]) |
| exact hp0.trans hq0.symm |
| have ht : 0<‖p‖ := norm_pos_iff.mpr hp0 |
| obtain ⟨a,ha,hpa⟩ := hp |
| let z : TangentBundle 𝓘(ℝ,Model n) M := ⟨x,‖p‖⁻¹ • p⟩ |
| let w : TangentBundle 𝓘(ℝ,Model n) M := ⟨x,‖p‖⁻¹ • q⟩ |
| have hzn : ‖z.2‖=1 := by |
| change ‖‖p‖⁻¹ • p‖=1 |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr ht),inv_mul_cancel₀ ht.ne'] |
| have hwn : ‖w.2‖=1 := by |
| change ‖‖p‖⁻¹ • q‖=1 |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr ht),←hn,inv_mul_cancel₀ ht.ne'] |
| have hflowz : (sprayFlow ‖p‖ z).1=riemannianExp x p := by |
| rw [←riemannianExp_smul,smul_smul,mul_inv_cancel₀ ht.ne',one_smul] |
| have hfloww : (sprayFlow ‖p‖ w).1=riemannianExp x q := by |
| rw [←riemannianExp_smul,smul_smul,mul_inv_cancel₀ ht.ne',one_smul] |
| have hflowa : (sprayFlow (a*‖p‖) z).1=riemannianExp x (a • p) := by |
| rw [←riemannianExp_smul,smul_smul,mul_assoc,mul_inv_cancel₀ ht.ne',mul_one] |
| have hzw := sprayFlow_unit_minimizer_unique z w rfl hzn hwn ht |
| (lt_mul_of_one_lt_left ht ha) |
| (by rw [hflowa]; exact hpa) |
| (by rw [hfloww,hn]; exact hq) |
| (by rw [hflowz,hfloww]; exact he) |
| have hvel : ‖p‖⁻¹ • p = ‖p‖⁻¹ • q := TotalSpace.mk_injective x hzw |
| exact (smul_right_inj (inv_ne_zero ht.ne')).mp hvel |
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| lemma riemannianExp_injOn_injectivityDomain (x : M) : |
| Set.InjOn (riemannianExp (n := n) x) (injectivityDomain x) := by |
| intro p hp q hq he |
| exact riemannianExp_interior_unique hp (injectivityDomain_subset_minimizingVectors x hq) he |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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