| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.NormalInverseVecNorm |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff |
| namespace YauCounterexamples |
|
|
| lemma compact_normal_image_neighborhood |
| (g : SmoothMetric NormalWaveSpace NormalWaveSpace) |
| {K V : Set NormalWaveSpace} (hK : IsCompact K) (hV : IsOpen V) (hKV : K ⊆ V) : |
| ∃ r > 0, ∀ q ∈ metricFrameSet g K, ∀ x : Fin 3 → ℝ, ‖x‖ < r → |
| normalJetMap q.1 q.2 ((metricChristoffel g q.1).bilinearComp q.2 q.2) (normalWaveEquiv x) ∈ V := by |
| let F : NormalWaveParameter × (Fin 3 → ℝ) → NormalWaveSpace := fun w => |
| (normalFamilyTotal g (w.1,normalWaveEquiv w.2)).2 |
| have hF : Continuous F := ((normalFamilyTotal_smooth g).continuous.comp |
| (continuous_fst.prodMk (normalWaveEquiv.continuous.comp continuous_snd))).snd |
| have hprod : metricFrameSet g K ×ˢ ({0} : Set (Fin 3 → ℝ)) ⊆ F ⁻¹' V := by |
| rintro ⟨q,x⟩ ⟨hq,hx⟩ |
| obtain rfl := Set.mem_singleton_iff.mp hx |
| change normalJetMap q.1 q.2 ((metricChristoffel g q.1).bilinearComp q.2 q.2) (normalWaveEquiv 0) ∈ V |
| rw [map_zero,normalJetMap_zero] |
| exact hKV hq.1 |
| obtain ⟨U,W,hU,hW,hKU,h0W,hUW⟩ := generalized_tube_lemma (metricFrameSet_isCompact g hK) |
| isCompact_singleton (hV.preimage hF) hprod |
| obtain ⟨r,hr,hrW⟩ := Metric.isOpen_iff.mp hW 0 (h0W (mem_singleton 0)) |
| refine ⟨r,hr,?_⟩ |
| intro q hq x hx |
| have hxW : x ∈ W := hrW (by simpa only [Metric.mem_ball,dist_zero_right] using hx) |
| have ht : (q,x) ∈ F ⁻¹' V := hUW ⟨hKU hq,hxW⟩ |
| exact ht |
| end YauCounterexamples |
| end |
|
|
| end OAI |
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