| import OAI.Geometry.NodalSets.Charts.MetricConnectionFamilyLemmas |
| import OAI.Geometry.NodalSets.Elliptic.TaylorPolynomial |
| import OAI.Geometry.NodalSets.Spectral.OrthonormalMetricFrame |
| |
| namespace OAI |
|
|
| namespace Yau.Geometry |
| open scoped ContDiff |
| open Yau.Jets |
| noncomputable section |
| attribute [local instance] clmTopology clmAdd clmModule |
|
|
| def frameConnection (B : Coord →L[ℝ] Coord →L[ℝ] Coord) (e : Coord ≃L[ℝ] Coord) : |
| Coord →L[ℝ] Coord →L[ℝ] Coord := |
| (((ContinuousLinearMap.compL ℝ Coord Coord Coord).flip e.toContinuousLinearMap).comp B).comp |
| e.toContinuousLinearMap |
|
|
| lemma frameConnection_apply (B : Coord →L[ℝ] Coord →L[ℝ] Coord) |
| (e : Coord ≃L[ℝ] Coord) (u v : Coord) : frameConnection B e u v = B (e u) (e v) := rfl |
|
|
| theorem positive_coordinate_frame (g : Coord →L[ℝ] Coord →L[ℝ] ℝ) |
| (hp : ∀ v : Coord, v ≠ 0 → 0 < g v v) (hs : ∀ u v, g u v = g v u) : |
| ∃ e : Coord ≃L[ℝ] Coord, ∀ i j : Fin 4, |
| g (e (Pi.single i 1)) (e (Pi.single j 1)) = if i = j then 1 else 0 := by |
| obtain ⟨b, hb⟩ : ∃ b : Module.Basis (Fin 4) ℝ Coord, |
| ∀ i j, g (b i) (b j) = if i = j then 1 else 0 := by |
| have hdim : Module.finrank ℝ Coord = 4 := Module.finrank_fin_fun (R := ℝ) |
| have h := positive_metric_orthonormal_basis g hp hs |
| rw [hdim] at h |
| exact h |
| refine ⟨b.equivFunL.symm, ?_⟩ |
| intro i j |
| have hei (i : Fin 4) : b.equivFunL.symm (Pi.single i 1) = b i := by |
| apply b.equivFunL.injective |
| rw [b.equivFunL.apply_symm_apply] |
| ext k |
| simp [Module.Basis.equivFunL, Module.Basis.equivFun, Pi.single_apply, Finsupp.single_apply, eq_comm] |
| rw [hei, hei] |
| exact hb i j |
| |
| theorem exists_orthonormal_normal_chart |
| (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) |
| (hg : ContDiff ℝ ∞ g) (hs : ∀ x u v, g x u v = g x v u) |
| (y : Coord) (hp : ∀ v : Coord, v ≠ 0 → 0 < g y v v) : |
| ∃ F : OpenPartialHomeomorph Coord Coord, |
| 0 ∈ F.source ∧ F 0 = y ∧ ContDiff ℝ ∞ (F : Coord → Coord) ∧ |
| ContDiffAt ℝ ∞ F.symm y ∧ |
| (∀ i j : Fin 4, pullbackMetric g F 0 (Pi.single i 1) (Pi.single j 1) = |
| if i = j then 1 else 0) ∧ |
| ∀ u v, fderiv ℝ (fun x ↦ pullbackMetric g F x u v) 0 = 0 := by |
| have hd := (hg.differentiable (by simp)).differentiableAt.hasFDerivAt (x := y) |
| obtain ⟨B, hB, hc⟩ := positive_metric_connection (g y) hp (hs y) (fderiv ℝ g y) |
| (metric_derivative_symmetric g (fderiv ℝ g y) y hd hs) |
| obtain ⟨e, he⟩ := positive_coordinate_frame (g y) hp (hs y) |
| let C := frameConnection B e |
| have hC : ∀ u v, C u v = C v u := fun u v ↦ hB (e u) (e v) |
| refine ⟨quadraticChart y e C, quadraticChart_source y e C, |
| quadraticChartMap_zero y e C, quadraticChartMap_smooth y e C, |
| quadraticChart_inverse_smooth y e C, ?_, ?_⟩ |
| · intro i j |
| rw [quadraticChart_coe, pullbackMetric_quadratic_zero] |
| exact he i j |
| · intro u v |
| exact quadratic_chart_metric_first_zero g (fderiv ℝ g y) y hd e C hC |
| (fun w u v ↦ hc (e w) (e u) (e v)) u v |
|
|
| end |
| end Yau.Geometry |
|
|
| end OAI |
|
|
