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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | 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
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