| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Coordinates.FixedJoinSmooth | |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.EigenOrder | |
| import OAI.Analysis.BiholderTransport.Regularity.FixedJoinMiddle | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Module | |
| open scoped Topology | |
| namespace WeakMTWTransport | |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] | |
| [FiniteDimensional ℝ E] {n : ℕ} | |
| lemma symmetric_eigenvalue_pos {A : E →ₗ[ℝ] E} (hA : A.IsSymmetric) | |
| (hn : finrank ℝ E=n) (hpos : ∀ x:E, x≠0 → 0 < inner ℝ (A x) x) (i:Fin n) : | |
| 0 < hA.eigenvalues hn i := by | |
| have H := hpos (hA.eigenvectorBasis hn i) ((hA.eigenvectorBasis hn).toBasis.ne_zero i) | |
| rw [hA.apply_eigenvectorBasis,real_inner_smul_left,real_inner_self_eq_norm_sq, | |
| (hA.eigenvectorBasis hn).norm_eq_one] at H | |
| simpa using H | |
| end WeakMTWTransport | |
| end | |
| noncomputable section | |
| open Set Filter Manifold Bundle Module | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| local instance tangentFiniteJoin (x : M) : FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| def fixedJoinOperator (x : M) (h T : ℝ) (p : TangentSpace 𝓘(ℝ,Model n) x) : | |
| TangentSpace 𝓘(ℝ,Model n) x →L[ℝ] TangentSpace 𝓘(ℝ,Model n) x := by | |
| let : FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := | |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) | |
| exact bilinearOperator (fixedJoinMiddle x h T p) | |
| @[simp] lemma fixedJoinOperator_inner (x : M) (h T : ℝ) | |
| (p d k : TangentSpace 𝓘(ℝ,Model n) x) : | |
| inner ℝ (fixedJoinOperator x h T p d) k=fixedJoinMiddle x h T p d k := | |
| bilinearOperator_inner _ d k | |
| lemma fixedJoinOperator_symmetric {x:M} {h T:ℝ} {p : TangentSpace 𝓘(ℝ,Model n) x} | |
| (hB : ContDiffAt ℝ 2 (fixedJoinAction x h T p) (0,p)) : | |
| (fixedJoinOperator x h T p).toLinearMap.IsSymmetric := | |
| bilinearOperator_symmetric (fixedJoinMiddle_symmetric hB) | |
| variable [CompactSpace M] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| lemma fixedJoinAction_shortened_contDiffAt {x : M} {h T:ℝ} | |
| {p : TangentSpace 𝓘(ℝ,Model n) x} (hh1 : h < 1) (hT : h < T) (hT1 : T < 1) | |
| (hleft : h • p∈injectivityDomain x) | |
| (hright : (1-h) • (sprayFlow h (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2∈ | |
| injectivityDomain (sprayFlow h (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1) : | |
| ContDiffAt ℝ ∞ (fixedJoinAction x h T p) (0,p) := by | |
| have hden : 0 < 1-h := sub_pos.mpr hh1 | |
| have ht0 : 0 < (T-h)/(1-h) := div_pos (sub_pos.mpr hT) hden | |
| have ht1 : (T-h)/(1-h) < 1 := (div_lt_one hden).mpr (by linarith) | |
| have H := contracted_minimizer_mem_injectivityDomain | |
| (injectivityDomain_subset_minimizingVectors _ hright) ht0 ht1 | |
| rw [smul_smul,div_mul_cancel₀ _ hden.ne'] at H | |
| exact fixedJoinAction_contDiffAt hT.ne' hleft H | |
| end WeakMTWTransport | |
| end | |
| end OAI | |