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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 | import OAI.Geometry.IsometricImmersion.Calculus.HessianCommutator
import OAI.Geometry.IsometricImmersion.Curvature.GaussConnection
import OAI.Geometry.IsometricImmersion.Energy.HeightEnergy
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
namespace OAI
noncomputable section
open scoped ContDiff Topology BigOperators Matrix
open Filter
namespace SmoothLocal.Geometry
variable {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord}
theorem metric_compatibility (hg : SmoothPositiveOn g U) (hU : IsOpen U)
(hp : p ∈ U) (k i j : Fin 2) :
coordPartial k (fun q => g q i j) p =
(∑ r, christoffel g r k i p * g p r j) +
(∑ r, christoffel g r k j p * g p i r) := by
have hlower (a b c : Fin 2) :
(∑ r, christoffel g r a b p * g p c r) = christoffelFirstKind g a b p c := by
simpa only [Matrix.mulVec, dotProduct, mul_comm] using
congrFun (metric_mul_christoffel hg hp a b) c
have hsym : (∑ r, christoffel g r k i p * g p r j) =
∑ r, christoffel g r k i p * g p j r := by
apply Finset.sum_congr rfl
intro r hr
rw [metric_coeff_symm hg hp r j]
rw [hsym, hlower k i j, hlower k j i]
simp only [christoffelFirstKind]
rw [metric_coeff_partial_symm hg hU hp k j i]
ring
theorem metricDet_coordPartial (hg : SmoothPositiveOn g U) (hU : IsOpen U)
(hp : p ∈ U) (k : Fin 2) :
coordPartial k (fun q => (g q).det) p =
coordPartial k (fun q => g q 0 0) p * g p 1 1 +
g p 0 0 * coordPartial k (fun q => g q 1 1) p -
(coordPartial k (fun q => g q 0 1) p * g p 1 0 +
g p 0 1 * coordPartial k (fun q => g q 1 0) p) := by
have hd (i j : Fin 2) : DifferentiableAt ℝ (fun q => g q i j) p :=
(((hg.1 i j) p hp).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp)
simp only [Matrix.det_fin_two]
have hprod₀ : DifferentiableAt ℝ (fun q => g q 0 0 * g q 1 1) p :=
(hd 0 0).mul (hd 1 1)
have hprod₁ : DifferentiableAt ℝ (fun q => g q 0 1 * g q 1 0) p :=
(hd 0 1).mul (hd 1 0)
rw [HessianCalculus.coordPartial_sub_at hprod₀ hprod₁ k,
HessianCalculus.coordPartial_mul_at (hd 0 0) (hd 1 1) k,
HessianCalculus.coordPartial_mul_at (hd 0 1) (hd 1 0) k]
theorem metricDet_partial_trace (hg : SmoothPositiveOn g U) (hU : IsOpen U)
(hp : p ∈ U) (k : Fin 2) :
coordPartial k (fun q => (g q).det) p =
2 * (christoffel g 0 k 0 p + christoffel g 1 k 1 p) * (g p).det := by
rw [metricDet_coordPartial hg hU hp k]
simp only [metric_compatibility hg hU hp, Fin.sum_univ_two, Matrix.det_fin_two]
ring
theorem metricDet_partial_y (hg : SmoothPositiveOn g U) (hU : IsOpen U)
(hp : p ∈ U) :
coordPartial 1 (fun q => (g q).det) p =
2 * (christoffel g 0 0 1 p + christoffel g 1 1 1 p) * (g p).det := by
rw [metricDet_partial_trace hg hU hp 1,
christoffel_lower_symm hg hU hp 0 1 0]
theorem heightEnergy_polynomial_symm (z : Coord → ℝ)
(hg : SmoothPositiveOn g U) (hp : p ∈ U) :
heightEnergy g z p = (g p).det -
(g p 1 1 * (coordPartial 0 z p) ^ 2 -
2 * g p 0 1 * coordPartial 0 z p * coordPartial 1 z p +
g p 0 0 * (coordPartial 1 z p) ^ 2) := by
rw [heightEnergy_polynomial z hg hp, metric_coeff_symm hg hp 1 0]
ring
namespace CompatibilityCalculus
theorem coordPartial_quadratic
{A B C X Y : Coord → ℝ} {p : Coord}
(hA : DifferentiableAt ℝ A p) (hB : DifferentiableAt ℝ B p)
(hC : DifferentiableAt ℝ C p) (hX : DifferentiableAt ℝ X p)
(hY : DifferentiableAt ℝ Y p) (k : Fin 2) :
coordPartial k (fun q => A q * (X q) ^ 2 -
2 * B q * X q * Y q + C q * (Y q) ^ 2) p =
coordPartial k A p * (X p) ^ 2 + 2 * A p * X p * coordPartial k X p -
(2 * coordPartial k B p * X p * Y p +
2 * B p * coordPartial k X p * Y p +
2 * B p * X p * coordPartial k Y p) +
coordPartial k C p * (Y p) ^ 2 + 2 * C p * Y p * coordPartial k Y p := by
have hd := ((hA.hasFDerivAt.fun_mul (hX.hasFDerivAt.fun_mul hX.hasFDerivAt)).sub
(((hB.hasFDerivAt.const_mul (2 : ℝ)).fun_mul hX.hasFDerivAt).fun_mul hY.hasFDerivAt)).add
(hC.hasFDerivAt.fun_mul (hY.hasFDerivAt.fun_mul hY.hasFDerivAt))
simp only [Pi.add_def, Pi.sub_def] at hd
simp only [pow_two, coordPartial]
rw [hd.fderiv]
simp only [add_apply, sub_apply, smul_apply, smul_eq_mul]
ring
end CompatibilityCalculus
theorem coordPartial_heightEnergy_polynomial
(hg : SmoothPositiveOn g U) (hU : IsOpen U)
(hz : ContDiffOn ℝ ∞ z U) (hp : p ∈ U) (k : Fin 2) :
coordPartial k (heightEnergy g z) p =
coordPartial k (fun q => (g q).det) p -
(coordPartial k (fun q => g q 1 1) p * (coordPartial 0 z p) ^ 2 +
2 * g p 1 1 * coordPartial 0 z p * coordPartial k (coordPartial 0 z) p -
(2 * coordPartial k (fun q => g q 0 1) p * coordPartial 0 z p * coordPartial 1 z p +
2 * g p 0 1 * coordPartial k (coordPartial 0 z) p * coordPartial 1 z p +
2 * g p 0 1 * coordPartial 0 z p * coordPartial k (coordPartial 1 z) p) +
coordPartial k (fun q => g q 0 0) p * (coordPartial 1 z p) ^ 2 +
2 * g p 0 0 * coordPartial 1 z p * coordPartial k (coordPartial 1 z) p) := by
have heq : heightEnergy g z =ᶠ[𝓝 p] (fun q => (g q).det -
(g q 1 1 * (coordPartial 0 z q) ^ 2 -
2 * g q 0 1 * coordPartial 0 z q * coordPartial 1 z q +
g q 0 0 * (coordPartial 1 z q) ^ 2)) := by
filter_upwards [hU.mem_nhds hp] with q hq
exact heightEnergy_polynomial_symm z hg hq
have hdg (i j : Fin 2) : DifferentiableAt ℝ (fun q => g q i j) p :=
(((hg.1 i j) p hp).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp)
have hdz (i : Fin 2) : DifferentiableAt ℝ (coordPartial i z) p :=
(((partial_contDiffOn hz hU i) p hp).contDiffAt
(hU.mem_nhds hp)).differentiableAt (by simp)
have hdet : DifferentiableAt ℝ (fun q => (g q).det) p :=
(((metricDet_contDiffOn hg) p hp).contDiffAt
(hU.mem_nhds hp)).differentiableAt (by simp)
have hquad : DifferentiableAt ℝ (fun q =>
g q 1 1 * (coordPartial 0 z q) ^ 2 -
2 * g q 0 1 * coordPartial 0 z q * coordPartial 1 z q +
g q 0 0 * (coordPartial 1 z q) ^ 2) p :=
(((hdg 1 1).mul ((hdz 0).pow 2)).sub
((((hdg 0 1).const_mul (2 : ℝ)).mul (hdz 0)).mul (hdz 1))).add
((hdg 0 0).mul ((hdz 1).pow 2))
change fderiv ℝ (heightEnergy g z) p (Pi.single k 1) = _
rw [heq.fderiv_eq]
change coordPartial k (fun q => (g q).det -
(g q 1 1 * (coordPartial 0 z q) ^ 2 -
2 * g q 0 1 * coordPartial 0 z q * coordPartial 1 z q +
g q 0 0 * (coordPartial 1 z q) ^ 2)) p = _
rw [HessianCalculus.coordPartial_sub_at hdet hquad k,
CompatibilityCalculus.coordPartial_quadratic
(hdg 1 1) (hdg 0 1) (hdg 0 0) (hdz 0) (hdz 1) k]
theorem coordPartial_heightEnergy
(hg : SmoothPositiveOn g U) (hU : IsOpen U)
(hz : ContDiffOn ℝ ∞ z U) (hp : p ∈ U) (k : Fin 2) :
coordPartial k (heightEnergy g z) p =
2 * (christoffel g 0 k 0 p + christoffel g 1 k 1 p) * heightEnergy g z p -
2 * (covHessian g z p k 0 *
(g p 1 1 * coordPartial 0 z p - g p 0 1 * coordPartial 1 z p) +
covHessian g z p k 1 *
(g p 0 0 * coordPartial 1 z p - g p 0 1 * coordPartial 0 z p)) := by
rw [coordPartial_heightEnergy_polynomial hg hU hz hp k,
metricDet_partial_trace hg hU hp k, heightEnergy_polynomial_symm z hg hp]
simp only [metric_compatibility hg hU hp, covHessian, Fin.sum_univ_two,
Matrix.det_fin_two]
rw [metric_coeff_symm hg hp 1 0]
ring
theorem heightEnergy_y_quotient
(hg : SmoothPositiveOn g U) (hU : IsOpen U)
(hz : ContDiffOn ℝ ∞ z U) (hp : p ∈ U)
(hyy : covHessian g z p 1 1 ≠ 0) :
coordPartial 1 (heightEnergy g z) p =
2 * (christoffel g 0 0 1 p + christoffel g 1 1 1 p) * heightEnergy g z p -
2 * covHessian g z p 1 1 *
((g p 0 0 - (covHessian g z p 0 1 / covHessian g z p 1 1) * g p 0 1) *
coordPartial 1 z p -
(g p 0 1 - (covHessian g z p 0 1 / covHessian g z p 1 1) * g p 1 1) *
coordPartial 0 z p) := by
rw [coordPartial_heightEnergy hg hU hz hp 1,
christoffel_lower_symm hg hU hp 0 1 0,
covHessian_symm hg hU hz hp 1 0]
field_simp [hyy]
ring
end SmoothLocal.Geometry
end
end OAI
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