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110f4f3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | import Mathlib
import OAI.Analysis.BiholderTransport.Regularity.FrameMetric
import OAI.Analysis.BiholderTransport.Coordinates.NormalFrameTaylor
import OAI.Analysis.BiholderTransport.Calculus.JetTranslation
namespace OAI
section
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)
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
end
end
end OAI
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