| import OAI.Geometry.Immersion.ClosedSurface.TensorCoordinates |
| |
| namespace OAI |
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| 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 |
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| 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) |
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| 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 |
| smul_eq_mul, RingHom.id_apply] |
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| def complexPullbackField (χ : Base → Base) (p : Base) : ComplexTensor →L[ℂ] ComplexTensor := |
| complexPullback (fderiv ℝ χ p) |
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| 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 |
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| 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 |
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| 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 |
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| lemma complexPullback_id (A : ComplexTensor) : |
| complexPullback (ContinuousLinearMap.id ℝ Base) A = A := by |
| ext i |
| fin_cases i < |
| secondDirection, dx, dy] |
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| 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] |
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| 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] |
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| 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) |
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| end ClosedSurfaceR4.PhaseMean |
| |
| namespace ClosedSurfaceR4.PhaseMean |
| open ClosedSurfaceR4.SmallModes (Base Field coordDeriv) |
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| 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 |
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| 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)) |
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| def coordinateTarget (e : Base → Base) (A : Base → ComplexTensor) : Base → ComplexTensor := |
| fun p => complexPullbackField e p (A (e p)) |
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| 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 |
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| end ClosedSurfaceR4.PhaseMean |
| |
| end |
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| end OAI |
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