| import OAI.Geometry.IsometricImmersion.Metrics.MetricSmoothness |
| import OAI.Geometry.IsometricImmersion.Immersions.InducedMetric |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped ContDiff Topology BigOperators Matrix |
| open Filter |
|
|
| namespace SmoothLocal.Geometry |
|
|
| namespace HessianCalculus |
|
|
| variable {f h : Coord → ℝ} {p : Coord} |
|
|
| theorem coordPartial_add_at |
| (hf : DifferentiableAt ℝ f p) (hh : DifferentiableAt ℝ h p) (i : Fin 2) : |
| coordPartial i (fun q => f q + h q) p = coordPartial i f p + coordPartial i h p := by |
| simp [coordPartial, fderiv_fun_add hf hh] |
|
|
| theorem coordPartial_sub_at |
| (hf : DifferentiableAt ℝ f p) (hh : DifferentiableAt ℝ h p) (i : Fin 2) : |
| coordPartial i (fun q => f q - h q) p = coordPartial i f p - coordPartial i h p := by |
| simp [coordPartial, fderiv_fun_sub hf hh] |
|
|
| theorem coordPartial_mul_at |
| (hf : DifferentiableAt ℝ f p) (hh : DifferentiableAt ℝ h p) (i : Fin 2) : |
| coordPartial i (fun q => f q * h q) p = |
| coordPartial i f p * h p + f p * coordPartial i h p := by |
| simp [coordPartial, fderiv_fun_mul hf hh, add_comm, mul_comm] |
|
|
| theorem coordPartial_sum_two (f : Fin 2 → Coord → ℝ) |
| (hf : ∀ a, DifferentiableAt ℝ (f a) p) (i : Fin 2) : |
| coordPartial i (fun q => ∑ a, f a q) p = ∑ a, coordPartial i (f a) p := by |
| simp only [Fin.sum_univ_two] |
| exact coordPartial_add_at (hf 0) (hf 1) i |
|
|
| end HessianCalculus |
|
|
| variable {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord} |
|
|
| theorem metric_coeff_symm (hg : SmoothPositiveOn g U) (hp : p ∈ U) (i j : Fin 2) : |
| g p i j = g p j i := by |
| simpa using (hg.2 p hp).1.apply j i |
|
|
| theorem metric_coeff_partial_symm (hg : SmoothPositiveOn g U) (hU : IsOpen U) |
| (hp : p ∈ U) (k i j : Fin 2) : |
| coordPartial k (fun q => g q i j) p = coordPartial k (fun q => g q j i) p := by |
| have heq : (fun q => g q i j) =ᶠ[𝓝 p] (fun q => g q j i) := by |
| filter_upwards [hU.mem_nhds hp] with q hq |
| exact metric_coeff_symm hg hq i j |
| unfold coordPartial |
| rw [heq.fderiv_eq] |
|
|
| theorem christoffel_lower_symm (hg : SmoothPositiveOn g U) (hU : IsOpen U) |
| (hp : p ∈ U) (k i j : Fin 2) : |
| christoffel g k i j p = christoffel g k j i p := by |
| unfold christoffel |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro l hl |
| rw [metric_coeff_partial_symm hg hU hp l i j] |
| ring |
|
|
| theorem covHessian_symm (hg : SmoothPositiveOn g U) (hU : IsOpen U) |
| (hz : ContDiffOn ℝ ∞ z U) (hp : p ∈ U) (i j : Fin 2) : |
| covHessian g z p i j = covHessian g z p j i := by |
| unfold covHessian |
| rw [coordPartial_comm hz hU hp i j] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro a ha |
| rw [christoffel_lower_symm hg hU hp a i j] |
|
|
| theorem coordPartial_covHessian (hg : SmoothPositiveOn g U) (hU : IsOpen U) |
| (hz : ContDiffOn ℝ ∞ z U) (hp : p ∈ U) (i j k : Fin 2) : |
| coordPartial i (fun q => covHessian g z q j k) p = |
| coordPartial i (coordPartial j (coordPartial k z)) p - |
| ∑ a, (coordPartial i (christoffel g a j k) p * coordPartial a z p + |
| christoffel g a j k p * coordPartial i (coordPartial a z) p) := by |
| have hz1 (a : Fin 2) : DifferentiableAt ℝ (coordPartial a z) p := |
| (((partial_contDiffOn hz hU a) p hp).contDiffAt |
| (hU.mem_nhds hp)).differentiableAt (by simp) |
| have hg1 (a : Fin 2) : DifferentiableAt ℝ (christoffel g a j k) p := |
| (((christoffel_contDiffOn hg hU a j k) p hp).contDiffAt |
| (hU.mem_nhds hp)).differentiableAt (by simp) |
| have hz2 : DifferentiableAt ℝ (coordPartial j (coordPartial k z)) p := |
| (((partial_contDiffOn (partial_contDiffOn hz hU k) hU j) p hp).contDiffAt |
| (hU.mem_nhds hp)).differentiableAt (by simp) |
| have hprod (a : Fin 2) : DifferentiableAt ℝ |
| (fun q => christoffel g a j k q * coordPartial a z q) p := |
| (hg1 a).mul (hz1 a) |
| have hsum : DifferentiableAt ℝ |
| (fun q => ∑ a, christoffel g a j k q * coordPartial a z q) p := by |
| exact DifferentiableAt.fun_sum fun a _ => hprod a |
| change coordPartial i (fun q => coordPartial j (coordPartial k z) q - |
| ∑ a, christoffel g a j k q * coordPartial a z q) p = _ |
| rw [HessianCalculus.coordPartial_sub_at hz2 hsum i, |
| HessianCalculus.coordPartial_sum_two |
| (fun a q => christoffel g a j k q * coordPartial a z q) hprod i] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro a ha |
| exact HessianCalculus.coordPartial_mul_at (hg1 a) (hz1 a) i |
|
|
| def covDerivHessian (g : MetricField) (z : Coord → ℝ) |
| (i j k : Fin 2) (p : Coord) : ℝ := |
| coordPartial i (fun q => covHessian g z q j k) p - |
| (∑ a, christoffel g a i j p * covHessian g z p a k) - |
| (∑ a, christoffel g a i k p * covHessian g z p j a) |
|
|
| theorem covHessian_commutator (hg : SmoothPositiveOn g U) (hU : IsOpen U) |
| (hz : ContDiffOn ℝ ∞ z U) (hp : p ∈ U) (i j k : Fin 2) : |
| covDerivHessian g z i j k p - covDerivHessian g z j i k p = |
| -(∑ a, riemann g a k i j p * coordPartial a z p) := by |
| have hGamma (a : Fin 2) : christoffel g a i j p = christoffel g a j i p := |
| christoffel_lower_symm hg hU hp a i j |
| unfold covDerivHessian |
| rw [coordPartial_covHessian hg hU hz hp i j k, |
| coordPartial_covHessian hg hU hz hp j i k, |
| coordPartial_comm (partial_contDiffOn hz hU k) hU hp i j] |
| simp only [covHessian, riemann, Fin.sum_univ_two] |
| rw [hGamma 0, hGamma 1] |
| ring |
|
|
| end SmoothLocal.Geometry |
|
|
| end |
|
|
| end OAI |
|
|
