On the Betti and Tachibana numbers of compact Einstein manifolds

TL;DR

This paper establishes new curvature-topology relationships for compact Einstein manifolds, showing conditions under which such manifolds are spherical space forms or relate to Tachibana numbers, advancing understanding of geometric structures.

Contribution

The paper proves that compact Einstein manifolds with positive Einstein constant are spherical space forms under certain curvature bounds and explores properties of Tachibana numbers for negative Einstein constant cases.

Findings

Compact Einstein manifolds with positive Einstein constant are spherical space forms if sectional curvature exceeds a specific bound.

Conditions are identified under which Tachibana numbers relate to the topology of Einstein manifolds.

The results connect curvature bounds with topological classifications of 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 $\alpha < 0$.