| import Mathlib |
| 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] |
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| 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) |
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|
| 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 |
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|
| 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)) |
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|
| 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] |
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|
| 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] |
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|
| 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 |
|
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| end ModelTangentJet |
| end WeakMTWTransport |
|
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| end |
| end |
|
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| end OAI |
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