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import OAI.Analysis.BiholderTransport.Contact.CoordinateScalarSupport
import OAI.Analysis.BiholderTransport.Regularity.CenterFullPullback
namespace OAI
section
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff
namespace WeakMTWTransport
section ModelTangentJet
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]
omit [CompactSpace M] [Nonempty 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] in
lemma center_frame_identity (a : M) (d : Model n) :
(trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ a d=d := by
rw [TangentBundle.symmL_trivializationAt_eq_core (mem_chart_source (Model n) a)]
exact tangentCoordChange_self (I := 𝓘(ℝ,Model n)) (mem_extChartAt_source a)
omit [CompactSpace M] [Nonempty M]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M] in
lemma model_tangent_second_fderiv {a : M} {f : TangentSpace 𝓘(ℝ,Model n) a → ℝ}
{q : Model n} (hf : ContDiffAt ℝ 2 f q) (d e : Model n) :
fderiv ℝ (fderiv ℝ (fun r : Model n=>f r)) q d e=
fderiv ℝ (fderiv ℝ f) q d e := by
let A := (trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ a
have hA (r : Model n) : A r=r := center_frame_identity a r
have hz (r : Model n) : (0 : TangentSpace 𝓘(ℝ,Model n) a)+A r=r :=
(zero_add (A r)).trans (hA r)
have H := second_fderiv_comp_affine (E := Model n) (F := TangentSpace 𝓘(ℝ,Model n) a) (f := f) A 0 q
(by simpa only [hz] using hf) d e
have he : (fun h : Model n => f ((0 : TangentSpace 𝓘(ℝ,Model n) a)+A h))=
(fun h : Model n => f h) := by funext h; rw [hz h]
rw [he,hz q,hA d,hA e] at H
exact H
omit [Nonempty M] in
lemma scalarCostSupport_endpoint_contDiffAt {a c : M}
{q : TangentSpace 𝓘(ℝ,Model n) a} {r : TangentSpace 𝓘(ℝ,Model n) c}
{v s : ℝ} {φ : ℝ → ℝ} (hr : s • r∈injectivityDomain c)
(he : riemannianExp a q=c) (hφ : ContDiffAt ℝ 2 φ v) :
ContDiffAt ℝ 2 (fun w=>scalarCostSupport φ v c r s (riemannianExp a w)) q := by
have hC := cost_contMDiffAt_of_injectivityDomain
(⟨c,s • r⟩ : TangentBundle 𝓘(ℝ,Model n) M) hr
have hC' : ContMDiffAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) 𝓘(ℝ,ℝ) ∞
(fun z:M×M=>cost z.1 z.2) (riemannianExp a q,riemannianExp c (s • r)) := by
rwa [he]
have hpair : ContMDiffAt 𝓘(ℝ,TangentSpace 𝓘(ℝ,Model n) a)
(𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) ∞
(fun w : TangentSpace 𝓘(ℝ,Model n) a => (riemannianExp a w,riemannianExp c (s • r))) q :=
(contMDiff_riemannianExp_fiber a q).prodMk contMDiffAt_const
have hcost : ContDiffAt ℝ ∞ (fun w : TangentSpace 𝓘(ℝ,Model n) a =>
cost (riemannianExp a w) (riemannianExp c (s • r))) q :=
(hC'.comp (g := fun z:M×M=>cost z.1 z.2) q hpair).contDiffAt
have hf : ContDiffAt ℝ ∞ (fun w=>v+cost c (riemannianExp c (s • r))/s-
cost (riemannianExp a w) (riemannianExp c (s • r))/s) q :=
contDiffAt_const.sub (hcost.div_const s)
have hval : v+cost c (riemannianExp c (s • r))/s-cost (riemannianExp a q) (riemannianExp c (s • r))/s=v := by
rw [he,add_sub_cancel_right]
exact (hval.symm ▸ hφ).comp q (hf.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2))
omit [Nonempty M] in
lemma model_riemannianExp_continuous (a : M) :
Continuous (fun w : Model n=>riemannianExp a w) := by
let A := (trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ a
have H := (continuous_riemannianExp a).comp A.continuous
have he : riemannianExp a ∘ A=(fun w : Model n=>riemannianExp a w) := by
funext w
exact congrArg (riemannianExp a) (center_frame_identity a w)
rwa [he] at H
omit [CompactSpace M] [Nonempty M]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M] in
lemma coordinateScalarSupport_center_value (c : M) (v s : ℝ) (φ : ℝ → ℝ)
(r z : Model n) :
coordinateScalarSupport c φ (((v,s),(extChartAt 𝓘(ℝ,Model n) c c,r)),z)=
scalarCostSupport φ v c r s ((extChartAt 𝓘(ℝ,Model n) c).symm z) := by
have hx := (extChartAt 𝓘(ℝ,Model n) c).map_source (mem_extChartAt_source c)
have hb := (extChartAt 𝓘(ℝ,Model n) c).left_inv (mem_extChartAt_source c)
rw [coordinateScalarSupport_eq hx,hb,center_frame_identity]
omit [Nonempty M] in
lemma coordinate_support_endpoint_germ {a c : M} {q r : Model n}
(v s : ℝ) (φ : ℝ → ℝ) (he : riemannianExp a q=c) :
(fun w : Model n=>coordinateScalarSupport c φ
(((v,s),(extChartAt 𝓘(ℝ,Model n) c c,r)),extChartAt 𝓘(ℝ,Model n) c (riemannianExp a w)))=ᶠ[𝓝 q]
(fun w : Model n=>scalarCostSupport φ v c r s (riemannianExp a w)) := by
have hx := (extChartAt 𝓘(ℝ,Model n) c).map_source (mem_extChartAt_source c)
have hb := (extChartAt 𝓘(ℝ,Model n) c).left_inv (mem_extChartAt_source c)
have hExp := (model_riemannianExp_continuous a).continuousAt (x := q)
have hqc : riemannianExp a q∈(extChartAt 𝓘(ℝ,Model n) c).source := by
rw [he]
exact mem_extChartAt_source c
have hsource := hExp.eventually ((isOpen_extChartAt_source c).mem_nhds hqc)
filter_upwards [hsource] with w hw
rw [coordinateScalarSupport_center_value,(extChartAt 𝓘(ℝ,Model n) c).left_inv hw]
omit [Nonempty M] in
lemma coordinate_support_endpoint_hessian {a c : M} {q r : Model n}
{v s : ℝ} {φ : ℝ → ℝ}
(hr : s • (show TangentSpace 𝓘(ℝ,Model n) c from r)∈injectivityDomain c)
(he : riemannianExp a q=c) (hφ : ContDiffAt ℝ 2 φ v) (d e : Model n) :
fderiv ℝ (fderiv ℝ (fun w : Model n=>coordinateScalarSupport c φ
(((v,s),(extChartAt 𝓘(ℝ,Model n) c c,r)),extChartAt 𝓘(ℝ,Model n) c (riemannianExp a w)))) q d e=
fderiv ℝ (fderiv ℝ (fun w : TangentSpace 𝓘(ℝ,Model n) a=>
scalarCostSupport φ v c r s (riemannianExp a w))) q d e := by
rw [(coordinate_support_endpoint_germ v s φ he).fderiv.fderiv_eq]
exact model_tangent_second_fderiv (scalarCostSupport_endpoint_contDiffAt hr he hφ) d e
omit [Nonempty M] in
lemma exists_uniform_coordinate_center_full_pullback :
∃ c0>0,∀ᶠ l : ℝ in 𝓝 1, 0<l → l<1 → ∀ a c : M,∀ r q : Model n,
(show TangentSpace 𝓘(ℝ,Model n) c from r)∈minimizingVectors c →
(show TangentSpace 𝓘(ℝ,Model n) a from q)∈injectivityDomain a →
riemannianExp c (l • r)=a → riemannianExp a q=c →
∀ (φ : ℝ → ℝ) (v : ℝ), ContDiffAt ℝ 2 φ v → deriv φ v=l → ∀ d : Model n,
c0*(1-l)*‖show TangentSpace 𝓘(ℝ,Model n) a from d‖^2+
(iteratedDeriv 2 φ v/l^2)*(inner ℝ (show TangentSpace 𝓘(ℝ,Model n) a from q) d)^2 ≤
inner ℝ (show TangentSpace 𝓘(ℝ,Model n) a from d) d+
fderiv ℝ (fderiv ℝ (fun w : Model n=>coordinateScalarSupport c φ
(((v,(1+l)/2),(extChartAt 𝓘(ℝ,Model n) c c,r)),
extChartAt 𝓘(ℝ,Model n) c (riemannianExp a w)))) q d d := by
obtain ⟨c0,hc0,H⟩ := exists_uniform_center_full_pullback (n := n) (M := M)
refine ⟨c0,hc0,?_⟩
filter_upwards [H] with l hl
intro hl0 hl1 a c r q hr hq hac hca φ v hφ hd d
rw [coordinate_support_endpoint_hessian
(contracted_minimizer_mem_injectivityDomain hr (by linarith) (by linarith)) hca hφ,
real_inner_self_eq_norm_sq]
exact hl hl0 hl1 c a r q hr hq hac hca φ v hφ hd d
end ModelTangentJet
end WeakMTWTransport
end
end
end OAI
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