| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.FamilyInverseFst |
| import OAI.Geometry.SmoothYau.Smoothness.NormalInverseVecNorm |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff |
| namespace YauCounterexamples |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] |
|
|
| theorem exists_centered_normal_inverse (g : SmoothMetric E E) {K : Set E} (hK : IsCompact K) : |
| ∃ rn > 0, ∃ r > 0, |
| ∃ e : (E × (E →L[ℝ] E)) → OpenPartialHomeomorph E E, |
| ∃ ψ : (E × (E →L[ℝ] E)) × E → E, |
| ContDiff ℝ ∞ ψ ∧ |
| (∀ q, (e q : E → E) = normalJetMap q.1 q.2 |
| ((metricChristoffel g q.1).bilinearComp q.2 q.2)) ∧ |
| (∀ q, ContDiffOn ℝ ∞ (e q).symm (e q).target) ∧ |
| (∀ q ∈ metricFrameSet g K, Metric.closedBall (0 : E) rn ⊆ (e q).source) ∧ |
| (∀ q ∈ metricFrameSet g K, ψ (q,0) = 0) ∧ |
| (∀ q ∈ metricFrameSet g K, ∀ t ∈ Metric.ball (0 : E) r, |
| q.1+t ∈ (e q).target ∧ (e q).symm (q.1+t) = ψ (q,t)) := by |
| obtain ⟨e,he,hs,hi⟩ := exists_normal_family_inverse g |
| obtain ⟨rn,hrn,hR⟩ := compact_normal_family_radius g hK |
| have hsub : metricFrameSet g K ×ˢ Metric.closedBall (0 : E) rn ⊆ e.source := by |
| rintro ⟨q,x⟩ ⟨hq,hx⟩ |
| rw [hs] |
| exact hR q hq x hx |
| let L := e '' (metricFrameSet g K ×ˢ Metric.closedBall (0 : E) rn) |
| have hL : IsCompact L := ((metricFrameSet_isCompact g hK).prod |
| (isCompact_closedBall 0 rn)).image_of_continuousOn (e.continuousOn.mono hsub) |
| have hLt : L ⊆ e.target := by |
| rintro _ ⟨w,hw,rfl⟩ |
| exact e.map_source (hsub hw) |
| obtain ⟨ψ,hψ,hEq⟩ := smooth_extension_near_compact hL e.open_target hLt hi.snd |
| obtain ⟨O,hO,hLO,hOeq⟩ := mem_nhdsSet_iff_exists.mp hEq |
| let shift : ((E × (E →L[ℝ] E)) × E) → ((E × (E →L[ℝ] E)) × E) := |
| fun w => (w.1,w.1.1+w.2) |
| have hshift : ContDiff ℝ ∞ shift := contDiff_fst.prodMk ((contDiff_fst.fst).add contDiff_snd) |
| have h₀ (q) (hq : q ∈ metricFrameSet g K) : (q,q.1) ∈ L := by |
| refine ⟨(q,0),⟨hq,Metric.mem_closedBall_self hrn.le⟩,?_⟩ |
| rw [he] |
| exact Prod.ext rfl (normalJetMap_zero _ _ _) |
| let T := shift ⁻¹' (O ∩ e.target) |
| have hT : IsOpen T := (hO.inter e.open_target).preimage hshift.continuous |
| have hKT : metricFrameSet g K ×ˢ {(0 : E)} ⊆ T := by |
| rintro ⟨q,t⟩ ⟨hq,ht⟩ |
| have ht0 : t = 0 := ht |
| subst t |
| change (q,q.1+0) ∈ O ∩ e.target |
| rw [add_zero] |
| exact ⟨hLO (h₀ q hq),hLt (h₀ q hq)⟩ |
| obtain ⟨U,V,hU,hV,hKU,h0V,hUV⟩ := generalized_tube_lemma (metricFrameSet_isCompact g hK) |
| isCompact_singleton hT hKT |
| obtain ⟨r,hr,hrV⟩ := Metric.isOpen_iff.mp hV 0 (h0V (mem_singleton 0)) |
| have he1 : ∀ w, (e w).1 = w.1 := by intro w; rw [he]; rfl |
| let eq := fiberOpenPartialHomeomorph e he1 |
| let ψc := ψ ∘ shift |
| have heq (q : E × (E →L[ℝ] E)) : (eq q : E → E) = normalJetMap q.1 q.2 |
| ((metricChristoffel g q.1).bilinearComp q.2 q.2) := by |
| funext x |
| change (e (q,x)).2 = _ |
| rw [he] |
| rfl |
| have hnear (q) (hq : q ∈ metricFrameSet g K) (t) (ht : t ∈ Metric.ball (0 : E) r) : |
| q.1+t ∈ (eq q).target ∧ (eq q).symm (q.1+t) = ψc (q,t) := by |
| have hm : (q,t) ∈ T := hUV ⟨hKU hq,hrV ht⟩ |
| change (q,q.1+t) ∈ O ∩ e.target at hm |
| exact ⟨hm.2,hOeq hm.1⟩ |
| refine ⟨rn,hrn,r,hr,eq,ψc,hψ.comp hshift,heq, |
| (fun q => contDiffOn_fiberInverse e he1 q hi),(fun q hq x hx => hsub ⟨hq,hx⟩),?_,hnear⟩ |
| intro q hq |
| have hh := (hnear q hq 0 (Metric.mem_ball_self hr)).2 |
| rw [add_zero] at hh |
| have hx0 : (0 : E) ∈ (eq q).source := hsub ⟨hq,Metric.mem_closedBall_self hrn.le⟩ |
| have hz : eq q 0 = q.1 := by rw [heq,normalJetMap_zero] |
| rw [←hh,←hz,(eq q).left_inv hx0] |
| |
| end YauCounterexamples |
| end |
| |
| end OAI |
| |
