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

On the Betti and Tachibana numbers of compact Einstein manifolds