File size: 5,982 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | import OAI.Geometry.Immersion.ClosedSurface.TensorCoordinates
namespace OAI
noncomputable section
open Set Complex Bundle Manifold
open scoped ContDiff Matrix Topology Manifold BigOperators
namespace ClosedSurfaceR4.PhaseMean
open ClosedSurfaceR4.SmallModes (Base dx dy)
open ClosedSurfaceR4.QuadraticMean (realMode)
open ClosedSurfaceR4.RealModes (base_decomposition)
abbrev ComplexTensor := Fin 3 → ℂ
def complexEvaluate (A : ComplexTensor) (v w : Base) : ℂ :=
A 0 * (v.1 : ℂ) * (w.1 : ℂ) +
A 1 * ((v.1 : ℂ) * (w.2 : ℂ) + (v.2 : ℂ) * (w.1 : ℂ)) +
A 2 * (v.2 : ℂ) * (w.2 : ℂ)
def complexPullback (J : Base →L[ℝ] Base) : ComplexTensor →L[ℂ] ComplexTensor :=
LinearMap.toContinuousLinearMap {
toFun := fun A i => complexEvaluate A (J (firstDirection i)) (J (secondDirection i))
map_add' := by intro A B; ext i; simp only [complexEvaluate, Pi.add_apply]; ring
map_smul' := by intro c A; ext i; simp only [complexEvaluate, Pi.smul_apply,
smul_eq_mul, RingHom.id_apply]; ring }
def complexPullbackField (χ : Base → Base) (p : Base) : ComplexTensor →L[ℂ] ComplexTensor :=
complexPullback (fderiv ℝ χ p)
lemma complexPullback_apply (J : Base →L[ℝ] Base) (A : ComplexTensor) (i : Fin 3) :
complexPullback J A i = complexEvaluate A (J (firstDirection i)) (J (secondDirection i)) := rfl
lemma complexEvaluate_pullback (J : Base →L[ℝ] Base) (A : ComplexTensor) (v w : Base) :
complexEvaluate (complexPullback J A) v w = complexEvaluate A (J v) (J w) := by
have hv : J v = v.1 • J dx + v.2 • J dy := by
conv_lhs => rw [base_decomposition v]
simp only [map_add, map_smul]
have hw : J w = w.1 • J dx + w.2 • J dy := by
conv_lhs => rw [base_decomposition w]
simp only [map_add, map_smul]
rw [hv, hw]
simp only [complexEvaluate, complexPullback_apply, firstDirection, secondDirection,
Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val,
Prod.fst_add, Prod.snd_add, Prod.smul_fst, Prod.smul_snd, smul_eq_mul,
Complex.ofReal_add, Complex.ofReal_mul]
ring
lemma complexPullback_comp (J K : Base →L[ℝ] Base) (A : ComplexTensor) :
complexPullback J (complexPullback K A) = complexPullback (K.comp J) A := by
ext i
rw [complexPullback_apply, complexEvaluate_pullback, complexPullback_apply]
rfl
lemma complexPullback_id (A : ComplexTensor) :
complexPullback (ContinuousLinearMap.id ℝ Base) A = A := by
ext i
fin_cases i <;> simp [complexPullback_apply, complexEvaluate, firstDirection,
secondDirection, dx, dy]
lemma realPart_complexPullback (J : Base →L[ℝ] Base) (A : ComplexTensor) :
ClosedSurfaceR4.QuadraticMean.realPart (complexPullback J A) = pullback J (ClosedSurfaceR4.QuadraticMean.realPart A) := by
ext i
change (complexEvaluate A _ _).re = evaluate (ClosedSurfaceR4.QuadraticMean.realPart A) _ _
simp [complexEvaluate, evaluate, ClosedSurfaceR4.QuadraticMean.realPart, Complex.mul_re]
lemma realMode_complexPullback (J : Base →L[ℝ] Base) (A : ComplexTensor) (θ : ℝ) :
realMode θ (complexPullback J A) = pullback J (realMode θ A) := by
unfold realMode
rw [← map_smul, realPart_complexPullback]
lemma complexPullbackField_inverse {χ e : Base → Base} {p : Base}
(hχ : DifferentiableAt ℝ χ p) (he : DifferentiableAt ℝ e (χ p))
(hinv : e ∘ χ =ᶠ[nhds p] id) (A : ComplexTensor) :
complexPullbackField χ p (complexPullbackField e (χ p) A) = A := by
unfold complexPullbackField
rw [complexPullback_comp, ← fderiv_comp p he hχ,
hinv.fderiv_eq, fderiv_id, complexPullback_id]
lemma oscillating_pullback_inverse {χ e : Base → Base} {p : Base}
(hχ : DifferentiableAt ℝ χ p) (he : DifferentiableAt ℝ e (χ p))
(hinv : e ∘ χ =ᶠ[nhds p] id) (A : ComplexTensor) (θ : ℝ) :
pullbackField χ p (realMode θ (complexPullbackField e (χ p) A)) = realMode θ A := by
change pullback (fderiv ℝ χ p) _ = _
rw [← realMode_complexPullback]
exact congrArg (realMode θ) (complexPullbackField_inverse hχ he hinv A)
end ClosedSurfaceR4.PhaseMean
namespace ClosedSurfaceR4.PhaseMean
open ClosedSurfaceR4.SmallModes (Base Field coordDeriv)
lemma contDiffOn_complexEvaluate {U : Set Base} {A : Base → ComplexTensor}
{v w : Base → Base} (hA : ContDiffOn ℝ ∞ A U) (hv : ContDiffOn ℝ ∞ v U)
(hw : ContDiffOn ℝ ∞ w U) :
ContDiffOn ℝ ∞ (fun p => complexEvaluate (A p) (v p) (w p)) U := by
have hvc := Complex.ofRealCLM.contDiff.comp_contDiffOn hv.fst
have hvd := Complex.ofRealCLM.contDiff.comp_contDiffOn hv.snd
have hwc := Complex.ofRealCLM.contDiff.comp_contDiffOn hw.fst
have hwd := Complex.ofRealCLM.contDiff.comp_contDiffOn hw.snd
have h0 := ((contDiffOn_pi.mp hA 0).mul hvc).mul hwc
have h1 := (contDiffOn_pi.mp hA 1).mul ((hvc.mul hwd).add (hvd.mul hwc))
have h2 := ((contDiffOn_pi.mp hA 2).mul hvd).mul hwd
exact (h0.add h1).add h2
lemma contDiffOn_complexPullbackField_apply {U : Set Base} (hU : IsOpen U)
{A : Base → ComplexTensor} {e : Base → Base}
(hA : ContDiffOn ℝ ∞ A U) (he : ContDiffOn ℝ ∞ e U) :
ContDiffOn ℝ ∞ (fun p => complexPullbackField e p (A p)) U := by
apply contDiffOn_pi.mpr
intro i
exact contDiffOn_complexEvaluate hA
(ClosedSurfaceR4.SmallModes.contDiffOn_coordDeriv_vector hU he (firstDirection i))
(ClosedSurfaceR4.SmallModes.contDiffOn_coordDeriv_vector hU he (secondDirection i))
def coordinateTarget (e : Base → Base) (A : Base → ComplexTensor) : Base → ComplexTensor :=
fun p => complexPullbackField e p (A (e p))
lemma contDiffOn_coordinateTarget {U V : Set Base} (hV : IsOpen V)
{A : Base → ComplexTensor} {e : Base → Base}
(hA : ContDiffOn ℝ ∞ A U) (he : ContDiffOn ℝ ∞ e V) (heU : Set.MapsTo e V U) :
ContDiffOn ℝ ∞ (coordinateTarget e A) V :=
contDiffOn_complexPullbackField_apply hV (hA.comp he heU) he
end ClosedSurfaceR4.PhaseMean
end
end OAI
|