| import OAI.Geometry.IsometricImmersion.Metrics.MetricCompatibility |
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| namespace OAI |
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| noncomputable section |
| open scoped ContDiff Topology BigOperators Matrix |
| open Filter |
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| namespace SmoothLocal.Geometry |
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| namespace MetricJetCalculus |
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| theorem coordPartial_const_mul_at {f : Coord → ℝ} {p : Coord} |
| (hf : DifferentiableAt ℝ f p) (c : ℝ) (i : Fin 2) : |
| coordPartial i (fun q => c * f q) p = c * coordPartial i f p := by |
| simp [coordPartial, fderiv_const_mul hf] |
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| end MetricJetCalculus |
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| theorem metric_inverse_derivative_identity {g : MetricField} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) {p : Coord} (hp : p ∈ U) |
| (d i j : Fin 2) : |
| (∑ a, (coordPartial d (fun q => g q i a) p * inverseMetric g p a j + |
| g p i a * coordPartial d (fun q => inverseMetric g q a j) p)) = 0 := by |
| have hproduct : (fun q => ∑ a, g q i a * inverseMetric g q a j) =ᶠ[𝓝 p] |
| (fun _ => (1 : Matrix (Fin 2) (Fin 2) ℝ) i j) := by |
| filter_upwards [hU.mem_nhds hp] with q hq |
| exact congrFun (congrFun |
| (Matrix.mul_nonsing_inv (g q) ((Matrix.isUnit_iff_isUnit_det _).mp (hg.2 q hq).isUnit)) i) j |
| have hdg (a : Fin 2) : DifferentiableAt ℝ (fun q => g q i a) p := |
| (((hg.1 i a) p hp).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp) |
| have hdi (a : Fin 2) : DifferentiableAt ℝ (fun q => inverseMetric g q a j) p := |
| (((inverseMetric_contDiffOn hg a j) p hp).contDiffAt |
| (hU.mem_nhds hp)).differentiableAt (by simp) |
| have hzero : coordPartial d (fun q => ∑ a, g q i a * inverseMetric g q a j) p = 0 := by |
| unfold coordPartial |
| rw [hproduct.fderiv_eq] |
| simp |
| have hmul (a : Fin 2) : DifferentiableAt ℝ |
| (fun q => g q i a * inverseMetric g q a j) p := (hdg a).mul (hdi a) |
| rw [HessianCalculus.coordPartial_sum_two _ hmul d] at hzero |
| simpa only [HessianCalculus.coordPartial_mul_at (hdg _) (hdi _) d] using hzero |
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| theorem inverseMetric_coordPartial_eq_of_firstJet |
| {g h : MetricField} {U : Set Coord} {p : Coord} |
| (hg : SmoothPositiveOn g U) (hh : SmoothPositiveOn h U) (hU : IsOpen U) |
| (hp : p ∈ U) (hval : g p = h p) |
| (hfirst : ∀ d i j, coordPartial d (fun q => g q i j) p = |
| coordPartial d (fun q => h q i j) p) (d i j : Fin 2) : |
| coordPartial d (fun q => inverseMetric g q i j) p = |
| coordPartial d (fun q => inverseMetric h q i j) p := by |
| have hinv : inverseMetric g p = inverseMetric h p := by simp only [inverseMetric, hval] |
| have hvec : g p *ᵥ (fun a => coordPartial d (fun q => inverseMetric g q a j) p) = |
| g p *ᵥ (fun a => coordPartial d (fun q => inverseMetric h q a j) p) := by |
| ext a |
| have hleft := metric_inverse_derivative_identity hg hU hp d a j |
| have hright := metric_inverse_derivative_identity hh hU hp d a j |
| simp only [hval, hfirst, hinv, Fin.sum_univ_two] at hleft |
| simp only [Fin.sum_univ_two] at hright |
| simp only [Matrix.mulVec, dotProduct, Fin.sum_univ_two, hval] |
| linarith |
| exact congrFun (Matrix.mulVec_injective_of_isUnit (hg.2 p hp).isUnit hvec) i |
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| theorem christoffel_coordPartial_formula {g : MetricField} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) {p : Coord} (hp : p ∈ U) |
| (d k i j : Fin 2) : |
| coordPartial d (christoffel g k i j) p = (1 / 2 : ℝ) * ∑ l, |
| (coordPartial d (fun q => inverseMetric g q k l) p * |
| (coordPartial i (fun q => g q j l) p + coordPartial j (fun q => g q i l) p - |
| coordPartial l (fun q => g q i j) p) + |
| inverseMetric g p k l * |
| (coordPartial d (coordPartial i (fun q => g q j l)) p + |
| coordPartial d (coordPartial j (fun q => g q i l)) p - |
| coordPartial d (coordPartial l (fun q => g q i j)) p)) := by |
| have hdg (a b c : Fin 2) : DifferentiableAt ℝ (coordPartial a (fun q => g q b c)) p := |
| (((partial_contDiffOn (hg.1 b c) hU a) p hp).contDiffAt |
| (hU.mem_nhds hp)).differentiableAt (by simp) |
| have hdi (l : Fin 2) : DifferentiableAt ℝ (fun q => inverseMetric g q k l) p := |
| (((inverseMetric_contDiffOn hg k l) p hp).contDiffAt |
| (hU.mem_nhds hp)).differentiableAt (by simp) |
| let B : Fin 2 → Coord → ℝ := fun l q => |
| coordPartial i (fun q => g q j l) q + coordPartial j (fun q => g q i l) q - |
| coordPartial l (fun q => g q i j) q |
| have hB (l : Fin 2) : DifferentiableAt ℝ (B l) p := |
| ((hdg i j l).add (hdg j i l)).sub (hdg l i j) |
| have hDB (l : Fin 2) : coordPartial d (B l) p = |
| coordPartial d (coordPartial i (fun q => g q j l)) p + |
| coordPartial d (coordPartial j (fun q => g q i l)) p - |
| coordPartial d (coordPartial l (fun q => g q i j)) p := by |
| change coordPartial d (fun q => |
| (coordPartial i (fun q => g q j l) q + coordPartial j (fun q => g q i l) q) - |
| coordPartial l (fun q => g q i j) q) p = _ |
| have hadd : DifferentiableAt ℝ |
| (fun q => coordPartial i (fun q => g q j l) q + coordPartial j (fun q => g q i l) q) p := |
| (hdg i j l).add (hdg j i l) |
| rw [HessianCalculus.coordPartial_sub_at hadd (hdg l i j) d, |
| HessianCalculus.coordPartial_add_at (hdg i j l) (hdg j i l) d] |
| have hsum : DifferentiableAt ℝ (fun q => ∑ l, inverseMetric g q k l * B l q) p := |
| DifferentiableAt.fun_sum fun l _ => (hdi l).mul (hB l) |
| change coordPartial d (fun q => (1 / 2 : ℝ) * ∑ l, inverseMetric g q k l * B l q) p = _ |
| have hterm (l : Fin 2) : DifferentiableAt ℝ |
| (fun q => inverseMetric g q k l * B l q) p := (hdi l).mul (hB l) |
| rw [MetricJetCalculus.coordPartial_const_mul_at hsum _ d, |
| HessianCalculus.coordPartial_sum_two _ hterm d] |
| simp only [HessianCalculus.coordPartial_mul_at (hdi _) (hB _) d, hDB, B] |
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| theorem christoffel_eq_of_firstJet {g h : MetricField} {p : Coord} |
| (hval : g p = h p) |
| (hfirst : ∀ d i j, coordPartial d (fun q => g q i j) p = |
| coordPartial d (fun q => h q i j) p) (k i j : Fin 2) : |
| christoffel g k i j p = christoffel h k i j p := by |
| simp only [christoffel, inverseMetric, hval, hfirst] |
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| theorem christoffel_coordPartial_eq_of_twoJet |
| {g h : MetricField} {U : Set Coord} {p : Coord} |
| (hg : SmoothPositiveOn g U) (hh : SmoothPositiveOn h U) (hU : IsOpen U) |
| (hp : p ∈ U) (hval : g p = h p) |
| (hfirst : ∀ d i j, coordPartial d (fun q => g q i j) p = |
| coordPartial d (fun q => h q i j) p) |
| (hsecond : ∀ d e i j, coordPartial d (coordPartial e (fun q => g q i j)) p = |
| coordPartial d (coordPartial e (fun q => h q i j)) p) (d k i j : Fin 2) : |
| coordPartial d (christoffel g k i j) p = coordPartial d (christoffel h k i j) p := by |
| have hInvD := inverseMetric_coordPartial_eq_of_firstJet hg hh hU hp hval hfirst |
| simp only [inverseMetric] at hInvD |
| rw [christoffel_coordPartial_formula hg hU hp d k i j, |
| christoffel_coordPartial_formula hh hU hp d k i j] |
| simp only [hInvD, hfirst, hsecond, inverseMetric, hval] |
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| theorem gaussianCurvature_eq_of_twoJet |
| {g h : MetricField} {U : Set Coord} {p : Coord} |
| (hg : SmoothPositiveOn g U) (hh : SmoothPositiveOn h U) (hU : IsOpen U) |
| (hp : p ∈ U) (hval : g p = h p) |
| (hfirst : ∀ d i j, coordPartial d (fun q => g q i j) p = |
| coordPartial d (fun q => h q i j) p) |
| (hsecond : ∀ d e i j, coordPartial d (coordPartial e (fun q => g q i j)) p = |
| coordPartial d (coordPartial e (fun q => h q i j)) p) : |
| gaussianCurvature g p = gaussianCurvature h p := by |
| have hΓ := christoffel_eq_of_firstJet hval hfirst |
| have hDΓ := christoffel_coordPartial_eq_of_twoJet hg hh hU hp hval hfirst hsecond |
| simp only [gaussianCurvature, riemann, hΓ, hDΓ, hval] |
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| end SmoothLocal.Geometry |
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| end |
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| end OAI |
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