| import OAI.Geometry.IsometricImmersion.Metric |
| import Mathlib.Analysis.Calculus.ContDiff.Operations |
| import Mathlib.Tactic.FinCases |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped ContDiff BigOperators Matrix |
|
|
| namespace SmoothLocal.Geometry |
|
|
| variable {g : MetricField} {U : Set Coord} |
|
|
| theorem metricDet_ne_zero (hg : SmoothPositiveOn g U) {p : Coord} (hp : p ∈ U) : |
| (g p).det ≠ 0 := |
| ((Matrix.isUnit_iff_isUnit_det _).mp (hg.2 p hp).isUnit).ne_zero |
|
|
| theorem metricDet_contDiffOn (hg : SmoothPositiveOn g U) : |
| ContDiffOn ℝ ∞ (fun p => (g p).det) U := by |
| simpa only [Matrix.det_fin_two] using |
| ((hg.1 0 0).mul (hg.1 1 1)).sub ((hg.1 0 1).mul (hg.1 1 0)) |
|
|
| theorem metricAdjugate_contDiffOn (hg : SmoothPositiveOn g U) (i j : Fin 2) : |
| ContDiffOn ℝ ∞ (fun p => (g p).adjugate i j) U := by |
| fin_cases i <;> fin_cases j |
| · simpa [Matrix.adjugate_fin_two] using hg.1 1 1 |
| · simpa [Matrix.adjugate_fin_two] using (hg.1 0 1).neg |
| · simpa [Matrix.adjugate_fin_two] using (hg.1 1 0).neg |
| · simpa [Matrix.adjugate_fin_two] using hg.1 0 0 |
|
|
| theorem inverseMetric_contDiffOn (hg : SmoothPositiveOn g U) (i j : Fin 2) : |
| ContDiffOn ℝ ∞ (fun p => inverseMetric g p i j) U := by |
| have hi : ContDiffOn ℝ ∞ (fun p => ((g p).det)⁻¹) U := |
| (metricDet_contDiffOn hg).inv (fun p hp => metricDet_ne_zero hg hp) |
| have ha := metricAdjugate_contDiffOn hg i j |
| simpa [inverseMetric, Matrix.inv_def, Ring.inverse_eq_inv, smul_eq_mul] using hi.mul ha |
|
|
| theorem christoffel_contDiffOn (hg : SmoothPositiveOn g U) (hU : IsOpen U) |
| (k i j : Fin 2) : |
| ContDiffOn ℝ ∞ (christoffel g k i j) U := by |
| have hs : ContDiffOn ℝ ∞ (fun p => ∑ l, inverseMetric g p k l * |
| (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)) U := by |
| apply ContDiffOn.sum |
| intro l hl |
| exact (inverseMetric_contDiffOn hg k l).mul |
| (((partial_contDiffOn (hg.1 j l) hU i).add |
| (partial_contDiffOn (hg.1 i l) hU j)).sub |
| (partial_contDiffOn (hg.1 i j) hU l)) |
| exact contDiffOn_const.mul hs |
|
|
| theorem covHessian_contDiffOn (hg : SmoothPositiveOn g U) (hU : IsOpen U) |
| {z : Coord → ℝ} (hz : ContDiffOn ℝ ∞ z U) (i j : Fin 2) : |
| ContDiffOn ℝ ∞ (fun p => covHessian g z p i j) U := by |
| have hs : ContDiffOn ℝ ∞ (fun p => ∑ k, |
| christoffel g k i j p * coordPartial k z p) U := by |
| apply ContDiffOn.sum |
| intro k hk |
| exact (christoffel_contDiffOn hg hU k i j).mul (partial_contDiffOn hz hU k) |
| exact (partial_contDiffOn (partial_contDiffOn hz hU j) hU i).sub hs |
|
|
| theorem covectorNormSq_contDiffOn (hg : SmoothPositiveOn g U) (hU : IsOpen U) |
| {z : Coord → ℝ} (hz : ContDiffOn ℝ ∞ z U) : |
| ContDiffOn ℝ ∞ (covectorNormSq g z) U := by |
| apply ContDiffOn.sum |
| intro i hi |
| apply ContDiffOn.sum |
| intro j hj |
| exact ((inverseMetric_contDiffOn hg i j).mul (partial_contDiffOn hz hU i)).mul |
| (partial_contDiffOn hz hU j) |
|
|
| theorem riemann_contDiffOn (hg : SmoothPositiveOn g U) (hU : IsOpen U) |
| (l k i j : Fin 2) : |
| ContDiffOn ℝ ∞ (riemann g l k i j) U := by |
| have hs : ContDiffOn ℝ ∞ (fun p => ∑ m, |
| (christoffel g m j k p * christoffel g l i m p - |
| christoffel g m i k p * christoffel g l j m p)) U := by |
| apply ContDiffOn.sum |
| intro m hm |
| exact ((christoffel_contDiffOn hg hU m j k).mul |
| (christoffel_contDiffOn hg hU l i m)).sub |
| ((christoffel_contDiffOn hg hU m i k).mul |
| (christoffel_contDiffOn hg hU l j m)) |
| exact ((partial_contDiffOn (christoffel_contDiffOn hg hU l j k) hU i).sub |
| (partial_contDiffOn (christoffel_contDiffOn hg hU l i k) hU j)).add hs |
|
|
| theorem gaussianCurvature_contDiffOn (hg : SmoothPositiveOn g U) (hU : IsOpen U) : |
| ContDiffOn ℝ ∞ (gaussianCurvature g) U := by |
| have hn : ContDiffOn ℝ ∞ (fun p => ∑ l, g p 0 l * riemann g l 1 0 1 p) U := by |
| apply ContDiffOn.sum |
| intro l hl |
| exact (hg.1 0 l).mul (riemann_contDiffOn hg hU l 1 0 1) |
| exact hn.div (metricDet_contDiffOn hg) (fun p hp => metricDet_ne_zero hg hp) |
|
|
| end SmoothLocal.Geometry |
|
|
| end |
|
|
| end OAI |
|
|
