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695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 | 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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