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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | import OAI.Geometry.IsometricImmersion.Metrics.MetricCompatibility
namespace OAI
noncomputable section
open scoped ContDiff Topology BigOperators Matrix
open Filter
namespace SmoothLocal.Geometry
namespace MetricJetCalculus
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]
end MetricJetCalculus
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
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
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]
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]
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]
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]
end SmoothLocal.Geometry
end
end OAI
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