openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Obstructions /LocalAdmissibility.lean
| import OAI.Geometry.IsometricImmersion.Calculus.AffineHessian | |
| import OAI.Geometry.IsometricImmersion.Immersions.RankOneOrientation | |
| import OAI.Geometry.IsometricImmersion.Curvature.NegativeCurvatureHeight | |
| import OAI.Geometry.IsometricImmersion.Calculus.HessianCommutator | |
| namespace OAI | |
| noncomputable section | |
| open scoped ContDiff Topology BigOperators Matrix | |
| open Filter | |
| namespace SmoothLocal.Geometry | |
| theorem exists_open_admissibility_neighborhood | |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord} | |
| (hg : SmoothPositiveOn g U) (hz : ContDiffOn ℝ ∞ z U) (hU : IsOpen U) | |
| (hp : p ∈ U) (hE : 0 < heightEnergy g z p) | |
| (hyy : covHessian g z p 1 1 ≠ 0) | |
| (hq : |covHessian g z p 0 1 / covHessian g z p 1 1| < (1 / 100 : ℝ)) : | |
| ∃ V : Set Coord, IsOpen V ∧ p ∈ V ∧ V ⊆ U ∧ | |
| ∀ q ∈ V, 0 < heightEnergy g z q ∧ covHessian g z q 1 1 ≠ 0 ∧ | |
| |covHessian g z q 0 1 / covHessian g z q 1 1| < (1 / 100 : ℝ) := by | |
| have hEc : ContinuousAt (heightEnergy g z) p := | |
| (heightEnergy_contDiffOn hg hU hz).continuousOn.continuousAt (hU.mem_nhds hp) | |
| have hHc (i j : Fin 2) : ContinuousAt (fun q => covHessian g z q i j) p := | |
| (covHessian_contDiffOn hg hU hz i j).continuousOn.continuousAt (hU.mem_nhds hp) | |
| have hqc : ContinuousAt | |
| (fun q => |covHessian g z q 0 1 / covHessian g z q 1 1|) p := | |
| ((hHc 0 1).div (hHc 1 1) hyy).abs | |
| have hgood : {q | q ∈ U ∧ 0 < heightEnergy g z q ∧ | |
| covHessian g z q 1 1 ≠ 0 ∧ | |
| |covHessian g z q 0 1 / covHessian g z q 1 1| < (1 / 100 : ℝ)} ∈ 𝓝 p := by | |
| filter_upwards [hU.mem_nhds hp, | |
| continuousAt_const.eventually_lt hEc hE, | |
| (hHc 1 1).eventually_ne hyy, | |
| hqc.eventually_lt continuousAt_const hq] with q hqU hqE hqyy hqq | |
| exact ⟨hqU, hqE, hqyy, hqq⟩ | |
| obtain ⟨V, hVsub, hVo, hpV⟩ := mem_nhds_iff.mp hgood | |
| exact ⟨V, hVo, hpV, fun q hqV => (hVsub hqV).1, | |
| fun q hqV => (hVsub hqV).2⟩ | |
| theorem exists_open_admissibility_at_critical | |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord} | |
| (hg : SmoothPositiveOn g U) (hz : ContDiffOn ℝ ∞ z U) (hU : IsOpen U) | |
| (hp : p ∈ U) (hcrit : fderiv ℝ z p = 0) | |
| (hyy : covHessian g z p 1 1 ≠ 0) | |
| (hq : |covHessian g z p 0 1 / covHessian g z p 1 1| < (1 / 100 : ℝ)) : | |
| ∃ V : Set Coord, IsOpen V ∧ p ∈ V ∧ V ⊆ U ∧ | |
| ∀ q ∈ V, 0 < heightEnergy g z q ∧ covHessian g z q 1 1 ≠ 0 ∧ | |
| |covHessian g z q 0 1 / covHessian g z q 1 1| < (1 / 100 : ℝ) := by | |
| apply exists_open_admissibility_neighborhood hg hz hU hp _ hyy hq | |
| rw [heightEnergy_at_critical g z p hcrit] | |
| exact (hg.2 p hp).det_pos | |
| theorem exists_oriented_immersion_admissibility | |
| {g : MetricField} {F : Coord → Ambient} {U : Set Coord} {b : Coord} | |
| (hg : SmoothPositiveOn g U) (hF : IsometricOn g F U) (hU : IsOpen U) | |
| (hb : b ∈ U) (n : Ambient) (hn : IsUnitNormalAt F n b) | |
| (hrank : (secondFundamental F n b).rank = 1) : | |
| ∃ R ∈ orientationRotations, ∃ V : Set Coord, | |
| IsOpen V ∧ (0 : Coord) ∈ V ∧ V ⊆ affineCoordinates b R ⁻¹' U ∧ | |
| SmoothPositiveOn (affinePullbackMetric g b R) V ∧ | |
| IsometricOn (affinePullbackMetric g b R) (affinePullbackImmersion F b R) V ∧ | |
| ∀ p ∈ V, | |
| let gR := affinePullbackMetric g b R | |
| let zR := height (affinePullbackImmersion F b R) n | |
| (covHessian gR zR p).det = gaussianCurvature gR p * heightEnergy gR zR p ∧ | |
| 0 < heightEnergy gR zR p ∧ covHessian gR zR p 1 1 ≠ 0 ∧ | |
| |covHessian gR zR p 0 1 / covHessian gR zR p 1 1| ≤ (1 / 100 : ℝ) := by | |
| have hz := height_smooth hF.1 n | |
| have hnpoint := geometric_height_normal_point hF hU hb hn | |
| have hsym : (covHessian g (height F n) b)ᵀ = covHessian g (height F n) b := by | |
| ext i j | |
| exact covHessian_symm hg hU hz hb j i | |
| have hHrank : (covHessian g (height F n) b).rank = 1 := by | |
| rw [hnpoint.2] | |
| exact hrank | |
| obtain ⟨R, hRnet, hRyy, hRq⟩ := | |
| exists_orientationRotation_for_rank_one (covHessian g (height F n) b) hsym hHrank | |
| have hRinj := mulVec_injective_of_transpose_mul_eq_one R | |
| (orientationRotations_properties hRnet).1 | |
| let W := affineCoordinates b R ⁻¹' U | |
| let gR := affinePullbackMetric g b R | |
| let FR := affinePullbackImmersion F b R | |
| let zR := height FR n | |
| have hWo : IsOpen W := isOpen_affine_preimage hU b R | |
| have h0W : (0 : Coord) ∈ W := by | |
| change affineCoordinates b R 0 ∈ U | |
| simpa only [affineCoordinates_zero] using hb | |
| have hgR : SmoothPositiveOn gR W := affinePullbackMetric_smoothPositive hg b R hRinj | |
| have hFR : IsometricOn gR FR W := affinePullbackImmersion_isometric hF hU b R | |
| have hzR : ContDiffOn ℝ ∞ zR W := height_smooth hFR.1 n | |
| have hcritR : fderiv ℝ zR 0 = 0 := by | |
| change fderiv ℝ ((height F n) ∘ affineCoordinates b R) 0 = 0 | |
| apply fderiv_affine_comp_eq_zero b R 0 | |
| · simpa only [affineCoordinates_zero] using | |
| (((hz b hb).contDiffAt (hU.mem_nhds hb)).differentiableAt (by simp)) | |
| · simpa only [affineCoordinates_zero] using hnpoint.1 | |
| have hHR : covHessian gR zR 0 = Rᵀ * covHessian g (height F n) b * R := by | |
| change covHessian (affinePullbackMetric g b R) | |
| ((height F n) ∘ affineCoordinates b R) 0 = _ | |
| exact covHessian_affinePullback_at_zero g hz hU b R hb hnpoint.1 | |
| have hyyR : covHessian gR zR 0 1 1 ≠ 0 := by rw [hHR]; exact hRyy | |
| have hqR : |covHessian gR zR 0 0 1 / covHessian gR zR 0 1 1| < (1 / 100 : ℝ) := by | |
| rw [hHR] | |
| exact hRq | |
| obtain ⟨V, hVo, h0V, hVW, hVad⟩ := | |
| exists_open_admissibility_at_critical hgR hzR hWo h0W hcritR hyyR hqR | |
| refine ⟨R, hRnet, V, hVo, h0V, hVW, ?_, ?_, ?_⟩ | |
| · exact ⟨fun i j => (hgR.1 i j).mono hVW, fun p hp => hgR.2 p (hVW hp)⟩ | |
| · exact ⟨hFR.1.mono hVW, fun p hp => hFR.2 p (hVW hp)⟩ | |
| · intro p hp | |
| have heq := geometric_height_equation hgR hFR hWo n hn.1 (hVW hp) | |
| have had := hVad p hp | |
| refine ⟨?_, had.1, had.2.1, had.2.2.le⟩ | |
| simpa only [heightEnergy, mul_assoc] using heq | |
| end SmoothLocal.Geometry | |
| end | |
| end OAI | |