| import OAI.Geometry.IsometricImmersion.Curvature.MetricCurvatureJet |
| import Mathlib.Analysis.Calculus.ContDiff.Bounds |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped ContDiff Topology BigOperators Matrix |
|
|
| namespace SmoothLocal.Geometry |
|
|
| section Smoothness |
| variable {T : Type*} [NormedAddCommGroup T] [NormedSpace ℝ T] {t : T} |
| variable {G : T → MetricMatrix} {D : T → MetricFirstJet} {DD : T → MetricSecondJet} |
|
|
| theorem metricMatrix_inverse_contDiffAt |
| (hG : ∀ i j, ContDiffAt ℝ ∞ (fun s => G s i j) t) (hdet : (G t).det ≠ 0) |
| (i j : Fin 2) : ContDiffAt ℝ ∞ (fun s => (G s)⁻¹ i j) t := by |
| have hdetc : ContDiffAt ℝ ∞ (fun s => (G s).det) t := by |
| simp only [Matrix.det_fin_two] |
| exact ((hG 0 0).mul (hG 1 1)).sub ((hG 0 1).mul (hG 1 0)) |
| have hi := hdetc.inv hdet |
| have ha : ContDiffAt ℝ ∞ (fun s => (G s).adjugate i j) t := by |
| fin_cases i <;> fin_cases j |
| · simpa [Matrix.adjugate_fin_two] using hG 1 1 |
| · convert (hG 0 1).neg using 1 |
| funext s |
| simp [Matrix.adjugate_fin_two] |
| · convert (hG 1 0).neg using 1 |
| funext s |
| simp [Matrix.adjugate_fin_two] |
| · simpa [Matrix.adjugate_fin_two] using hG 0 0 |
| have hh : ContDiffAt ℝ ∞ (fun s => (G s).det⁻¹ * (G s).adjugate i j) t := hi.mul ha |
| simpa [Matrix.inv_def, Ring.inverse_eq_inv, smul_eq_mul] using hh |
|
|
| theorem inverseDerivativeJet_contDiffAt |
| (hG : ∀ i j, ContDiffAt ℝ ∞ (fun s => G s i j) t) |
| (hD : ∀ d i j, ContDiffAt ℝ ∞ (fun s => D s d i j) t) |
| (hdet : (G t).det ≠ 0) (d i j : Fin 2) : |
| ContDiffAt ℝ ∞ (fun s => inverseDerivativeJet (G s) (D s) d i j) t := by |
| have hi := metricMatrix_inverse_contDiffAt hG hdet |
| have ht (a b : Fin 2) : |
| ContDiffAt ℝ ∞ (fun s => (G s)⁻¹ i a * D s d a b * (G s)⁻¹ b j) t := |
| ((hi i a).mul (hD d a b)).mul (hi b j) |
| simpa only [inverseDerivativeJet, Fin.sum_univ_two] using |
| (((ht 0 0).add (ht 0 1)).add ((ht 1 0).add (ht 1 1))).neg |
|
|
| theorem christoffelJet_contDiffAt |
| (hG : ∀ i j, ContDiffAt ℝ ∞ (fun s => G s i j) t) |
| (hD : ∀ d i j, ContDiffAt ℝ ∞ (fun s => D s d i j) t) |
| (hdet : (G t).det ≠ 0) (k i j : Fin 2) : |
| ContDiffAt ℝ ∞ (fun s => christoffelJet (G s) (D s) k i j) t := by |
| have hi := metricMatrix_inverse_contDiffAt hG hdet |
| have ht (l : Fin 2) : ContDiffAt ℝ ∞ |
| (fun s => (G s)⁻¹ k l * (D s i j l + D s j i l - D s l i j)) t := |
| (hi k l).mul (((hD i j l).add (hD j i l)).sub (hD l i j)) |
| simpa only [christoffelJet, Fin.sum_univ_two] using |
| (contDiffAt_const (c := (1 / 2 : ℝ))).mul ((ht 0).add (ht 1)) |
|
|
| theorem christoffelDerivativeJet_contDiffAt |
| (hG : ∀ i j, ContDiffAt ℝ ∞ (fun s => G s i j) t) |
| (hD : ∀ d i j, ContDiffAt ℝ ∞ (fun s => D s d i j) t) |
| (hDD : ∀ d e i j, ContDiffAt ℝ ∞ (fun s => DD s d e i j) t) |
| (hdet : (G t).det ≠ 0) (d k i j : Fin 2) : |
| ContDiffAt ℝ ∞ (fun s => christoffelDerivativeJet (G s) (D s) (DD s) d k i j) t := by |
| have hi := metricMatrix_inverse_contDiffAt hG hdet |
| have hdi := inverseDerivativeJet_contDiffAt hG hD hdet |
| have ht (l : Fin 2) : ContDiffAt ℝ ∞ (fun s => |
| inverseDerivativeJet (G s) (D s) d k l * (D s i j l + D s j i l - D s l i j) + |
| (G s)⁻¹ k l * (DD s d i j l + DD s d j i l - DD s d l i j)) t := |
| ((hdi d k l).mul (((hD i j l).add (hD j i l)).sub (hD l i j))).add |
| ((hi k l).mul (((hDD d i j l).add (hDD d j i l)).sub (hDD d l i j))) |
| simpa only [christoffelDerivativeJet, Fin.sum_univ_two] using |
| (contDiffAt_const (c := (1 / 2 : ℝ))).mul ((ht 0).add (ht 1)) |
|
|
| theorem riemannJet_contDiffAt |
| (hG : ∀ i j, ContDiffAt ℝ ∞ (fun s => G s i j) t) |
| (hD : ∀ d i j, ContDiffAt ℝ ∞ (fun s => D s d i j) t) |
| (hDD : ∀ d e i j, ContDiffAt ℝ ∞ (fun s => DD s d e i j) t) |
| (hdet : (G t).det ≠ 0) (l k i j : Fin 2) : |
| ContDiffAt ℝ ∞ (fun s => riemannJet (G s) (D s) (DD s) l k i j) t := by |
| have hΓ := christoffelJet_contDiffAt hG hD hdet |
| have hDΓ := christoffelDerivativeJet_contDiffAt hG hD hDD hdet |
| have ht (m : Fin 2) : ContDiffAt ℝ ∞ (fun s => |
| christoffelJet (G s) (D s) m j k * christoffelJet (G s) (D s) l i m - |
| christoffelJet (G s) (D s) m i k * christoffelJet (G s) (D s) l j m) t := |
| ((hΓ m j k).mul (hΓ l i m)).sub ((hΓ m i k).mul (hΓ l j m)) |
| simpa only [riemannJet, Fin.sum_univ_two] using |
| ((hDΓ i l j k).sub (hDΓ j l i k)).add ((ht 0).add (ht 1)) |
|
|
| theorem curvatureJet_contDiffAt |
| (hG : ∀ i j, ContDiffAt ℝ ∞ (fun s => G s i j) t) |
| (hD : ∀ d i j, ContDiffAt ℝ ∞ (fun s => D s d i j) t) |
| (hDD : ∀ d e i j, ContDiffAt ℝ ∞ (fun s => DD s d e i j) t) |
| (hdet : (G t).det ≠ 0) : |
| ContDiffAt ℝ ∞ (fun s => curvatureJet (G s) (D s) (DD s)) t := by |
| have hR := riemannJet_contDiffAt hG hD hDD hdet |
| have hn : ContDiffAt ℝ ∞ (fun s => ∑ l, G s 0 l * riemannJet (G s) (D s) (DD s) l 1 0 1) t := by |
| simpa only [Fin.sum_univ_two] using |
| ((hG 0 0).mul (hR 0 1 0 1)).add ((hG 0 1).mul (hR 1 1 0 1)) |
| have hd : ContDiffAt ℝ ∞ (fun s => (G s).det) t := by |
| simp only [Matrix.det_fin_two] |
| exact ((hG 0 0).mul (hG 1 1)).sub ((hG 0 1).mul (hG 1 0)) |
| exact hn.div hd hdet |
|
|
| end Smoothness |
| end SmoothLocal.Geometry |
|
|
| end |
|
|
| end OAI |
|
|
