| import Mathlib |
| import OAI.Analysis.BiholderTransport.Regularity.FrameMetric |
| import OAI.Analysis.BiholderTransport.Coordinates.NormalFrameTaylor |
| import OAI.Analysis.BiholderTransport.Calculus.JetTranslation |
|
|
| namespace OAI |
|
|
| section |
|
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| noncomputable section |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
|
|
| namespace WeakMTWTransport |
| section ChartTrueMatrix |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty 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] |
|
|
| def chartJetMatrix (a c:M) (b q:Model n) (p:Model n →L[ℝ] ℝ) |
| (A:Model n →L[ℝ] Model n →L[ℝ] ℝ) : Model n →L[ℝ] Model n →L[ℝ] ℝ := |
| frameMetric a b+secondJetPullback p A (fderiv ℝ (movingPrefixChart a c 1 b) q) |
| (fderiv ℝ (fderiv ℝ (movingPrefixChart a c 1 b)) q) |
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| omit [Nonempty M] in |
| lemma chartJetMatrix_normal_identity {f:M → ℝ} {a c y:M} {b q:Model n} |
| (hb:b∈(extChartAt 𝓘(ℝ,Model n) a).target) |
| (hq:chartFiberInverse a b q∈injectivityDomain ((extChartAt 𝓘(ℝ,Model n) a).symm b)) |
| (hy:movingPrefix a 1 b q=y) (hc:y∈(extChartAt 𝓘(ℝ,Model n) c).source) |
| {r:TangentSpace 𝓘(ℝ,Model n) y} (hr:r∈injectivityDomain y) |
| (hrx:riemannianExp y r=(extChartAt 𝓘(ℝ,Model n) a).symm b) |
| {S:TangentSpace 𝓘(ℝ,Model n) y →L[ℝ] TangentSpace 𝓘(ℝ,Model n) y} |
| (hS:HasQuadraticExpansion (fun h=>f (riemannianExp y h)) r S) |
| {p:Model n} {A:Model n →L[ℝ] Model n} (hA:∀d e,inner ℝ (A d) e=inner ℝ d (A e)) |
| (hAe:HasQuadraticExpansion (fun h=>f ((extChartAt 𝓘(ℝ,Model n) c).symm |
| (extChartAt 𝓘(ℝ,Model n) c y+h))) p A) |
| {R:TangentSpace 𝓘(ℝ,Model n) ((extChartAt 𝓘(ℝ,Model n) a).symm b) → TangentSpace 𝓘(ℝ,Model n) y} |
| (hR:DifferentiableAt ℝ R (chartFiberInverse a b q)) (hR0:R (chartFiberInverse a b q)=0) |
| (hRe:∀ᶠ v in 𝓝 (chartFiberInverse a b q),riemannianExp y (R v)= |
| riemannianExp ((extChartAt 𝓘(ℝ,Model n) a).symm b) v) : |
| ∀d,chartJetMatrix a c b q (innerSL ℝ p) ((innerSL ℝ).comp A) d d= |
| inner ℝ (S (fderiv ℝ R (chartFiberInverse a b q) (chartFiberInverse a b d))) |
| (fderiv ℝ R (chartFiberInverse a b q) (chartFiberInverse a b d))+ |
| normalHessian y r (fderiv ℝ R (chartFiberInverse a b q) (chartFiberInverse a b d)) |
| (fderiv ℝ R (chartFiberInverse a b q) (chartFiberInverse a b d)) := by |
| let B:=chartFiberInverse a b |
| have he:riemannianExp ((extChartAt 𝓘(ℝ,Model n) a).symm b) (B q)=y:=by |
| simpa only [movingPrefix_eq hb,one_smul,B,chartFiberInverse] using hy |
| have H:=chart_normal_pole_frame_identity B hr hq hrx he hc hS hA hAe hR hR0 hRe |
| have hg:ContDiffAt ℝ 2 (movingPrefixChart a c 1 b) q:=by |
| exact ((movingPrefixChart_contDiffAt hb (hy.symm ▸ hc)).comp q |
| (contDiffAt_const.prodMk contDiffAt_id)).of_le |
| (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2) |
| have heq:(fun h=>movingPrefixChart a c 1 b (q+h)-extChartAt 𝓘(ℝ,Model n) c y)= |
| (fun h=>extChartAt 𝓘(ℝ,Model n) c (riemannianExp |
| ((extChartAt 𝓘(ℝ,Model n) a).symm b) (B q+B h))-extChartAt 𝓘(ℝ,Model n) c y):=by |
| funext h |
| simp only [movingPrefixChart,movingPrefix_eq hb,one_smul,map_add,B,chartFiberInverse] |
| rw [←heq] at H |
| dsimp only at H |
| rw [fderiv_translate_at_zero (hg.differentiableAt (by norm_num)), |
| second_fderiv_translate_at_zero hg] at H |
| intro d |
| rw [chartJetMatrix,add_apply,add_apply, |
| frameMetric_apply hb,real_inner_self_eq_norm_sq] |
| exact H d |
|
|
| omit [Nonempty M] in |
| lemma chartJetMatrix_tendsto {a c:M} {b q:ℕ → Model n} {b0 q0:Model n} |
| {p:ℕ → Model n →L[ℝ] ℝ} {p0:Model n →L[ℝ] ℝ} |
| {A:ℕ → Model n →L[ℝ] Model n →L[ℝ] ℝ} {A0:Model n →L[ℝ] Model n →L[ℝ] ℝ} |
| (hb0:b0∈(extChartAt 𝓘(ℝ,Model n) a).target) |
| (hc:movingPrefix a 1 b0 q0∈(extChartAt 𝓘(ℝ,Model n) c).source) |
| (hb:Tendsto b atTop (𝓝 b0)) (hq:Tendsto q atTop (𝓝 q0)) |
| (hp:Tendsto p atTop (𝓝 p0)) (hA:Tendsto A atTop (𝓝 A0)) : |
| Tendsto (fun k=>chartJetMatrix a c (b k) (q k) (p k) (A k)) atTop |
| (𝓝 (chartJetMatrix a c b0 q0 p0 A0)) := by |
| let : NormedAddCommGroup (Model n →L[ℝ] ℝ) := ContinuousLinearMap.toNormedAddCommGroup |
| let : NormedSpace ℝ (Model n →L[ℝ] ℝ) := ContinuousLinearMap.toNormedSpace |
| have hg:=movingPrefixChart_contDiffAt hb0 hc |
| have hbx:=hb.prodMk_nhds hq |
| have hD:=((ContDiffAt.partial_snd_fderiv hg).continuousAt.tendsto).comp hbx |
| have hD2:=((ContDiffAt.partial_snd_fderiv_two hg).continuousAt.tendsto).comp hbx |
| have hf : Tendsto (fun k=>frameMetric a (b k)) atTop (𝓝 (frameMetric a b0)) := |
| (frameMetric_continuousAt hb0).tendsto.comp hb |
| have hs : Tendsto (fun k=>secondJetPullback (p k) (A k) |
| (fderiv ℝ (movingPrefixChart a c 1 (b k)) (q k)) |
| (fderiv ℝ (fderiv ℝ (movingPrefixChart a c 1 (b k))) (q k))) atTop |
| (𝓝 (secondJetPullback p0 A0 (fderiv ℝ (movingPrefixChart a c 1 b0) q0) |
| (fderiv ℝ (fderiv ℝ (movingPrefixChart a c 1 b0)) q0))) := |
| secondJetPullback_tendsto hp hA hD hD2 |
| exact hf.add hs |
| end ChartTrueMatrix |
| end WeakMTWTransport |
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| end |
| end |
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| end OAI |
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