openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Obstructions /DiskBoundaryObstruction.lean
| import OAI.Geometry.IsometricImmersion.Coordinates.RoundDisk | |
| import OAI.Geometry.IsometricImmersion.Immersions.CompactBoundaryExtremum | |
| import OAI.Geometry.IsometricImmersion.Immersions.RankOne2 | |
| namespace OAI | |
| noncomputable section | |
| open Set | |
| open scoped ContDiff Topology Matrix | |
| namespace SmoothLocal.Geometry | |
| variable {g : MetricField} {F : Coord → Ambient} {U : Set Coord} | |
| variable {center : Coord} {radius : ℝ} | |
| theorem exists_roundDisk_boundary_nonzero_secondFundamental | |
| (hg : SmoothPositiveOn g U) (hF : IsometricOn g F U) (hU : IsOpen U) | |
| (hr : 0 < radius) (hDU : roundClosedDisk center radius ⊆ U) | |
| (hK : ∀ p ∈ roundOpenDisk center radius, gaussianCurvature g p < 0) : | |
| ∃ b ∈ frontier (roundClosedDisk center radius), | |
| secondFundamental F (unitNormalField F b) b ≠ 0 := by | |
| by_contra hnone | |
| have hcircle : ∀ t, coordinateCircle center radius t ∈ U := | |
| fun t => hDU (coordinateCircle_mem_roundClosedDisk center radius t) | |
| have hII : ∀ t, secondFundamental F | |
| (unitNormalField F (coordinateCircle center radius t)) | |
| (coordinateCircle center radius t) = 0 := by | |
| intro t | |
| by_contra ht | |
| exact hnone ⟨coordinateCircle center radius t, | |
| coordinateCircle_mem_frontier_roundClosedDisk center hr t, ht⟩ | |
| obtain ⟨e, c, he, _, _, hh⟩ := | |
| exists_circle_constant_normal_and_height hg hF hU hcircle hII | |
| have hK' : ∀ p ∈ interior (roundClosedDisk center radius), gaussianCurvature g p < 0 := by | |
| intro p hp | |
| exact hK p (by simpa only [interior_roundClosedDisk center hr] using hp) | |
| apply immersion_height_not_constant_frontier hg hF hU | |
| (isCompact_roundClosedDisk center hr) hDU | |
| (interior_roundClosedDisk_nonempty center hr) hK' e he | |
| refine ⟨c, ?_⟩ | |
| intro p hp | |
| obtain ⟨t, rfl⟩ := exists_coordinateCircle_of_mem_frontier_roundClosedDisk center p hr hp | |
| exact hh t | |
| theorem roundDisk_boundary_rank_one | |
| (hg : SmoothPositiveOn g U) (hF : IsometricOn g F U) (hU : IsOpen U) | |
| (hr : 0 < radius) (hDU : roundClosedDisk center radius ⊆ U) | |
| (hK : ∀ p ∈ roundOpenDisk center radius, gaussianCurvature g p < 0) | |
| (hKboundary : ∀ p ∈ frontier (roundClosedDisk center radius), gaussianCurvature g p = 0) : | |
| ∃ b ∈ frontier (roundClosedDisk center radius), | |
| secondFundamental F (unitNormalField F b) b ≠ 0 ∧ | |
| (secondFundamental F (unitNormalField F b) b).rank = 1 := by | |
| obtain ⟨b, hb, hII⟩ := exists_roundDisk_boundary_nonzero_secondFundamental | |
| hg hF hU hr hDU hK | |
| have hbD : b ∈ roundClosedDisk center radius := | |
| ((mem_frontier_roundClosedDisk_iff center b hr).mp hb).le | |
| have hbU : b ∈ U := hDU hbD | |
| have hdet := det_secondFundamental_eq_gaussianCurvature_mul_det hg hF hU hbU | |
| (unitNormalField_isUnitNormalAt hg hF hbU) | |
| rw [hKboundary b hb, zero_mul] at hdet | |
| exact ⟨b, hb, hII, two_by_two_rank_eq_one_of_ne_zero_det_eq_zero hII hdet⟩ | |
| theorem exists_roundDisk_boundary_unitNormal_rank_one | |
| (hg : SmoothPositiveOn g U) (hF : IsometricOn g F U) (hU : IsOpen U) | |
| (hr : 0 < radius) (hDU : roundClosedDisk center radius ⊆ U) | |
| (hK : ∀ p ∈ roundOpenDisk center radius, gaussianCurvature g p < 0) | |
| (hKboundary : ∀ p ∈ frontier (roundClosedDisk center radius), gaussianCurvature g p = 0) : | |
| ∃ b ∈ frontier (roundClosedDisk center radius), ∃ n : Ambient, | |
| IsUnitNormalAt F n b ∧ secondFundamental F n b ≠ 0 ∧ | |
| (secondFundamental F n b).rank = 1 := by | |
| obtain ⟨b, hb, hII, hrank⟩ := roundDisk_boundary_rank_one hg hF hU hr hDU hK hKboundary | |
| have hbU : b ∈ U := hDU (((mem_frontier_roundClosedDisk_iff center b hr).mp hb).le) | |
| exact ⟨b, hb, unitNormalField F b, unitNormalField_isUnitNormalAt hg hF hbU, hII, hrank⟩ | |
| end SmoothLocal.Geometry | |
| end | |
| end OAI | |