| import Mathlib |
| import OAI.Analysis.BiholderTransport.CostGeometry.BranchCost2 |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.ContactDifferential |
| import OAI.Analysis.BiholderTransport.Regularity.Reversal |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff |
|
|
| namespace WeakMTWTransport |
| section Diagonal |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
|
|
| lemma diagonal_stationary_hessian_identity {F : E×E → ℝ} {p : E} |
| (hF : ContDiffAt ℝ 2 F (p,p)) |
| (hstat : ∀ᶠ v in 𝓝 p, ∀ k, fderiv ℝ F (v,v) (0,k)=0) (a k : E) : |
| fderiv ℝ (fderiv ℝ F) (p,p) (0,a) (0,k)= |
| -fderiv ℝ (fderiv ℝ F) (p,p) (a,0) (0,k) := by |
| have hdf := (hF.fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num) |
| let L : E →L[ℝ] E×E := (ContinuousLinearMap.id ℝ E).prod (ContinuousLinearMap.id ℝ E) |
| have hD := ((hdf.hasFDerivAt.comp (f := fun v : E => (v,v)) p L.hasFDerivAt).clm_apply |
| (hasFDerivAt_const ((0:E),k) p)).fderiv |
| have heq : (fun v : E => fderiv ℝ F (v,v) (0,k)) =ᶠ[𝓝 p] (fun _ => (0:ℝ)) := |
| hstat.mono (fun _ h => h k) |
| have hz : fderiv ℝ (fun v : E => fderiv ℝ F (v,v) (0,k)) p=0 := by |
| rw [heq.fderiv_eq] |
| exact (hasFDerivAt_const (0:ℝ) p).fderiv |
| have H := congrArg (fun A : E →L[ℝ] ℝ => A a) (hD.symm.trans hz) |
| simp only [add_apply,ContinuousLinearMap.comp_apply,zero_apply,map_zero,zero_add, |
| ContinuousLinearMap.flip_apply,L,ContinuousLinearMap.prod_apply,ContinuousLinearMap.id_apply] at H |
| have hsplit : (a,a)=(a,0)+((0:E),a) := by ext <;> simp |
| rw [hsplit,map_add,add_apply] at H |
| linarith |
| end Diagonal |
| section SplitFactor |
| 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] |
|
|
| lemma diagonalSplitAction_hessian_identity {x : M} {p : TangentSpace 𝓘(ℝ,Model n) x} |
| {t : ℝ} (ht : 0<t) (ht1 : t<1) (hleft : t • p∈injectivityDomain x) |
| (hright : (1-t) • (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2∈ |
| injectivityDomain (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1) |
| (a k : TangentSpace 𝓘(ℝ,Model n) x) : |
| fderiv ℝ (fderiv ℝ (diagonalSplitAction x t)) (p,p) (0,a) (0,k)= |
| -fderiv ℝ (fderiv ℝ (diagonalSplitAction x t)) (p,p) (a,0) (0,k) := by |
| exact diagonal_stationary_hessian_identity |
| ((diagonalSplitAction_contDiffAt hleft hright).of_le |
| (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)) |
| ((split_regular_legs_near hleft hright).mono |
| (fun v hv k => diagonalSplitAction_stationary ht ht1 hv.1 hv.2 k)) a k |
|
|
| lemma exists_suffix_log_in_prefix_coordinates {x : M} |
| {p : TangentSpace 𝓘(ℝ,Model n) x} {t : ℝ} |
| (hright : (1-t) • (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2∈ |
| injectivityDomain (sprayFlow t (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1) : |
| ∃ q : TangentSpace 𝓘(ℝ,Model n) x → |
| TangentSpace 𝓘(ℝ,Model n) (riemannianExp x p), |
| ContDiffAt ℝ ∞ q p ∧ |
| (∀ᶠ v in 𝓝 p, q v∈injectivityDomain (riemannianExp x p)) ∧ |
| (∀ᶠ v in 𝓝 p, riemannianExp (riemannianExp x p) (q v)=riemannianExp x (t • v)) := by |
| let : Nonempty M := ⟨x⟩ |
| let : MeasurableSpace M := borel M |
| let : BorelSpace M := ⟨rfl⟩ |
| let z : TangentBundle 𝓘(ℝ,Model n) M := ⟨x,p⟩ |
| let s := tangentScale (1-t) (sprayFlow t z) |
| have hs : s.2∈injectivityDomain s.1 := hright |
| have hr := reverseRay_injectivityDomain hs |
| have he : (reverseRay s).1=riemannianExp x p := by |
| change (sprayFlow 1 s).1=riemannianExp x p |
| rw [←riemannianExp_eq_sprayFlow] |
| exact shifted_exp_endpoint x p t |
| have he' := reverseRay_endpoint s |
| change riemannianExp (reverseRay s).1 (reverseRay s).2=(sprayFlow t z).1 at he' |
| rw [←riemannianExp_smul] at he' |
| generalize hrdef : reverseRay s=r at hr he he' |
| rcases r with ⟨y,r⟩ |
| dsimp only at he hr he' |
| subst y |
| obtain ⟨κ,hκ0,hκ,hκright⟩ := exists_normal_local_inverse_at hr |
| rw [he'] at hκ0 hκ hκright |
| let q := fun v : TangentSpace 𝓘(ℝ,Model n) x => κ (riemannianExp x (t • v)) |
| have hE : ContMDiffAt 𝓘(ℝ,TangentSpace 𝓘(ℝ,Model n) x) 𝓘(ℝ,Model n) ∞ |
| (fun v => riemannianExp x (t • v)) p := |
| (contMDiff_riemannianExp_fiber x _).comp p (show ContDiffAt ℝ ∞ (fun v : TangentSpace 𝓘(ℝ,Model n) x => t • v) p from |
| by fun_prop).contMDiffAt |
| have hq : ContDiffAt ℝ ∞ q p := (hκ.comp p hE).contDiffAt |
| refine ⟨q,hq,?_,hE.continuousAt.eventually hκright⟩ |
| apply hq.continuousAt.preimage_mem_nhds |
| apply (isOpen_injectivityDomain _).mem_nhds |
| change κ (riemannianExp x (t • p))∈_ |
| rw [hκ0] |
| exact hr |
|
|
| end SplitFactor |
| end WeakMTWTransport |
|
|
| end |
|
|
| end |
|
|
| end |
|
|
| end OAI |
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|
