| import Mathlib |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.ActualRadial |
| import OAI.Analysis.BiholderTransport.Coordinates.Prefix |
|
|
| namespace OAI |
|
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| section |
|
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| section |
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|
| noncomputable section |
| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology BigOperators |
|
|
| namespace WeakMTWTransport |
|
|
| lemma symmetric_bilinear_eq_of_diagonal |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {A B : E →L[ℝ] E →L[ℝ] ℝ} |
| (hA : ∀ u v, A u v = A v u) (hB : ∀ u v, B u v = B v u) |
| (hd : ∀ u, A u u = B u u) : A = B := by |
| ext u v |
| have H := hd (u+v) |
| simp only [map_add,add_apply,hA v u,hB v u] at H |
| linarith [hd u,hd v] |
|
|
| section GeometryJets |
| 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 prefixNormalCost_source_hessian {x : M} {p : TangentSpace 𝓘(ℝ,Model n) x} |
| {t : ℝ} (hp : t • p ∈ injectivityDomain x) |
| (u : TangentSpace 𝓘(ℝ,Model n) x) : |
| fderiv ℝ (fderiv ℝ (prefixNormalCost x t)) (0,p) (u,0) (u,0) = |
| normalHessian x (t • p) u u/t := by |
| have H := iteratedDeriv_two_affine_line |
| ((prefixNormalCost_contDiffAt hp).of_le (m := 2) |
| (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)) (u,0) |
| simp only [prefixNormalCost,Prod.fst_add,Prod.smul_fst,Prod.snd_add,Prod.smul_snd, |
| zero_add,smul_zero,add_zero,iteratedDeriv_div_const] at H |
| rw [← H,← hessianValue_eq_normalHessian hp] |
| rfl |
| end GeometryJets |
| end WeakMTWTransport |
|
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| end |
|
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| end |
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| end |
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| end OAI |
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|
