On Einstein four-manifolds

Abstract

In this paper we obtain obstructions to the existence of Einstein metrics satisfying auxiliary sectional curvature bounds. In particular, we give sufficient conditions for a compact-oriented Einstein four-manifold M to be isometric to either the sphere S 4 or the complex projective space CP 2. Also, we improve the Hitchin–Thorpe’s inequality which relates the Euler characteristic of M and its signature.