| import OAI.Geometry.EinsteinFour.ActualModelIdentification | |
| namespace OAI | |
| namespace PositiveEinsteinFour | |
| open EinsteinFour | |
| open scoped Manifold ContDiff | |
| /-- Every closed connected Einstein four-manifold with strictly positive sectional | |
| curvature is isometric, after positive scaling, to the round sphere, complex | |
| projective plane, or real projective four-space. -/ | |
| theorem classification (M : Type*) [TopologicalSpace M] [ChartedSpace FourSpace M] | |
| [IsManifold I4 ∞ M] [T2Space M] [SecondCountableTopology M] | |
| [CompactSpace M] [ConnectedSpace M] (g : SmoothMetric M) | |
| (hEinstein : IsEinstein g) (hpositive : HasPositiveSectionalCurvature g) : | |
| ∃ a : ℝ, 0 < a ∧ | |
| (ScaledIsometricTo g a roundDistance ∨ | |
| ScaledIsometricTo g a fubiniStudyDistance ∨ | |
| ScaledIsometricTo g a realProjectiveDistance) := | |
| classificationOn M g hEinstein hpositive | |
| end PositiveEinsteinFour | |
| end OAI | |