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59d1c80 | 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 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | 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
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