| import OAI.Geometry.Immersion.ClosedSurface.ImmersionJets | |
| namespace OAI | |
| noncomputable section | |
| open Set Complex Bundle Manifold | |
| open scoped ContDiff Matrix Topology Manifold BigOperators | |
| namespace ClosedSurfaceR4.PhaseGeometry | |
| open Set Function Filter | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [FiniteDimensional ℝ E] {M : Type*} [TopologicalSpace M] | |
| [ChartedSpace E M] [IsManifold 𝓘(ℝ, E) ∞ M] [T2Space M] | |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| theorem smooth_extension_near {F : M → V} {U : Set M} (hU : IsOpen U) | |
| (hF : ContMDiffOn 𝓘(ℝ, E) 𝓘(ℝ, V) ∞ F U) {p : M} (hp : p ∈ U) : | |
| ∃ G : M → V, ContMDiff 𝓘(ℝ, E) 𝓘(ℝ, V) ∞ G ∧ | |
| HasCompactSupport G ∧ tsupport G ⊆ U ∧ G =ᶠ[𝓝 p] F := by | |
| obtain ⟨b, _, hb⟩ := (SmoothBumpFunction.nhds_basis_tsupport (I := 𝓘(ℝ, E)) p).mem_iff.mp | |
| (hU.mem_nhds hp) | |
| let G : M → V := fun q => b q • F q | |
| have hs : tsupport G ⊆ tsupport b := tsupport_smul_subset_left _ _ | |
| refine ⟨G, ?_, ?_, hs.trans hb, ?_⟩ | |
| · apply contMDiff_of_tsupport | |
| intro q hq | |
| exact b.contMDiffAt.smul ((hF q (hb (hs hq))).contMDiffAt | |
| (hU.mem_nhds (hb (hs hq)))) | |
| · exact b.hasCompactSupport.of_isClosed_subset (isClosed_tsupport G) hs | |
| · filter_upwards [b.eventuallyEq_one] with q hq | |
| simp only [G, hq, Pi.one_apply, one_smul] | |
| end ClosedSurfaceR4.PhaseGeometry | |
| namespace ClosedSurfaceR4.RealModes | |
| open ClosedSurfaceR4.SmallModes ClosedSurfaceR4.PhaseGeometry Set Filter | |
| lemma coordDeriv_eventuallyEq {n : ℕ} {F G : RField n} {p : Base} | |
| (h : F =ᶠ[𝓝 p] G) (v : Base) : coordDeriv v F =ᶠ[𝓝 p] coordDeriv v G := by | |
| filter_upwards [h.fderiv (𝕜 := ℝ)] with q hq | |
| exact congrArg (fun A : Base →L[ℝ] RVec n => A v) hq | |
| lemma realSecondForm_eventuallyEq {F G : RField 4} {p : Base} | |
| (h : F =ᶠ[𝓝 p] G) (v w : Base) : | |
| (fun q => realSecondForm F v w q) =ᶠ[𝓝 p] (fun q => realSecondForm G v w q) := by | |
| filter_upwards [coordDeriv_eventuallyEq h dx, coordDeriv_eventuallyEq h dy, | |
| coordDeriv_eventuallyEq (coordDeriv_eventuallyEq h w) v] with q hx hy hxy | |
| simp only [realSecondForm, hx, hy, hxy] | |
| lemma realSecondTensor_eventuallyEq {F G : RField 4} {p : Base} | |
| (h : F =ᶠ[𝓝 p] G) : realSecondTensor F =ᶠ[𝓝 p] realSecondTensor G := by | |
| filter_upwards [realSecondForm_eventuallyEq h dx dx, realSecondForm_eventuallyEq h dx dy, | |
| realSecondForm_eventuallyEq h dy dy] with q hxx hxy hyy | |
| ext i | |
| fin_cases i <;> simp [realSecondTensor, hxx, hxy, hyy] | |
| theorem actual_local_phase_data_on {F : RField 4} {O : Set Base} | |
| (hO : IsOpen O) (hF : ContDiffOn ℝ ∞ F O) | |
| {H : Base → PhaseMean.Tensor} {p : Base} (hp : p ∈ O) (hH : ContinuousAt H p) | |
| (hImm : Function.Injective (fderiv ℝ F p)) | |
| (hB : ∃ v w : Base, realSecondForm F v w p ≠ 0) | |
| (hH0 : 0 < H p 0) (hHdet : 0 < H p 0 * H p 2 - (H p 1)^2) : | |
| ∃ (G : RField 4) (ξ : Fin 3 → Base) | |
| (Q : Fin 3 → PhaseMean.Tensor →L[ℝ] ℝ) (U : Set Base), | |
| ContDiff ℝ ∞ G ∧ HasCompactSupport G ∧ | |
| IsOpen U ∧ p ∈ U ∧ U ⊆ O ∧ EqOn G F U ∧ | |
| (∀ A, ∑ i, Q i A • covectorSquare (ξ i) = A) ∧ | |
| (∀ x ∈ U, Function.Injective (fderiv ℝ F x) ∧ | |
| (∀ i, 0 < Q i (H x)) ∧ (∀ i, Good (realSecondTensor F x) (ξ i)) ∧ | |
| (∀ i j, i ≠ j → Good (realSecondTensor F x) (ξ i + ξ j) ∧ | |
| Good (realSecondTensor F x) (ξ i - ξ j))) := by | |
| obtain ⟨G,hG,hGc,hGsp,he⟩ := smooth_extension_near hO hF.contMDiffOn hp | |
| have heir := he.fderiv_eq (𝕜 := ℝ) | |
| have hGi : Function.Injective (fderiv ℝ G p) := by rw [heir]; exact hImm | |
| have hGB : ∃ v w : Base, realSecondForm G v w p ≠ 0 := by | |
| obtain ⟨v,w,hvw⟩ := hB | |
| exact ⟨v,w,by rwa [(realSecondForm_eventuallyEq he v w).eq_of_nhds]⟩ | |
| obtain ⟨ξ,Q,V,hV,hpV,hdec,hdata⟩ := actual_local_phase_data hG.contDiff hH hGi hGB hH0 hHdet | |
| have hnb : ∀ᶠ x in 𝓝 p, x ∈ O ∧ x ∈ V ∧ G x = F x ∧ | |
| fderiv ℝ G x = fderiv ℝ F x ∧ realSecondTensor G x = realSecondTensor F x := by | |
| filter_upwards [hO.mem_nhds hp, hV.mem_nhds hpV, he, he.fderiv (𝕜 := ℝ), | |
| realSecondTensor_eventuallyEq he] with x hxO hxV hx hd hb | |
| exact ⟨hxO,hxV,hx,hd,hb⟩ | |
| obtain ⟨U,hUs,hU,hpU⟩ := mem_nhds_iff.mp hnb | |
| refine ⟨G,ξ,Q,U,hG.contDiff,hGc,hU,hpU,fun x hx => (hUs hx).1, | |
| fun x hx => (hUs hx).2.2.1,hdec,?_⟩ | |
| intro x hx | |
| obtain ⟨hxO,hxV,he,hd,hb⟩ := hUs hx | |
| have hh := hdata x hxV | |
| rwa [hd,hb] at hh | |
| end ClosedSurfaceR4.RealModes | |
| end | |
| end OAI | |