Four-manifolds with positive curvature operator

Abstract

A compact surface with positive mean scalar curvature must be diffeomorphic to the sphere S 2 or the real projective space RP 2 .A compact three-manifold with positive Ricci curvature must be diffeomorphic to the sphere S 3 or a quotient of it by a finite group of fixed point free isometries in the standard metric, such as the real projective space RP 3 or a lens space L 3 p q .This was proven in [1].Our main result is the following generalization to four dimensions. Theorem.A compact four-manifold with a positive curvature operator is diffeomorphic to the sphere S 4 or the real projective space RP 4 .Here we regard the Riemannian curvature tensor Rm = {R iJkl } as a symmetric bilinear form on the two-forms Λ 2 by lettingWe say the manifold has a positive curvature operator if Rm(φ, φ) > 0 for all two-forms φ Φ 0, and a nonnegative curvature operator if Rm(φ,φ) ^ 0 for all φ.These results extend to the case of nonnegative curvature.A compact surface with nonnegative mean scalar curvature must be diffeomorphic to a quotient of the sphere S 2 or the plane R 2 by a group of fixed-point free isometries in the standard metrics.The examples are the sphere S 2 , the real projective space RP 2 , the torus T 2 = S ι X S\ and the Klein bottle(where # denotes the connected sum). Theorem.A compact three-manifold with nonnegative Ricci curvature is diffeomorphic to a quotient of one of the spaces S 3 or S 2 X R ι or R 3 by a group of fixed point free isometries in the standard metrics.The quotients of S 2 X R 1 include S 2 X S\ RP 2 X S\ the unoriented S 2 bundle over S\ and the connected sum K 3 = RP 3 #RP 3 .The quotients of R 3 are the torus T 3 and five other flat three-manifolds.

Locations

• Journal of Differential Geometry - View - PDF