On the Betti and Tachibana numbers of compact Einstein manifolds

Abstract

Throughout the history of Einstein manifolds, differential geometers have shown great interest in finding the relationships between curvature and the topology of Einstein manifolds. In the paper, first, we prove that a compact Einstein manifold $(M,g)$ with Einstein constant $\alpha >0$ is a homo-logical sphere when the minimum of its sectional curvatures $> \alpha/(n+ 2)$; in particular, $(M,g)$ is a spherical space form when the minimum of its sectional curvatures $> \alpha / n$. Second, we prove two propositions (similar to the above ones) for Tachibana numbers of a compact Einstein manifold $(M,g)$ with $\alpha < 0$.