Diameter, volume, and topology for positive Ricci curvature
The author establishes the following result: Let \(M^ n\) be a compact Riemannian manifold with Ricci curvature \(\geq 1\). There is a number \(\epsilon >0\) such that for \(diam(M)>\pi -\epsilon\), \(M^ n\) is homeomorphic to the unit n-sphere. We remark that \(\epsilon\) depends on n, on a lower bound of the sectional curvature of \(M^ n\) and on a lower bound of the injectivity radius. The lemma used to prove the previous result allows to deduce a known theorem: If \(M^ n\) is a compact Riemannian manifold with sectional curvature \(\geq 1\) and \(diam(M)>\pi /2\), then \(M^ n\) is homeomorphic to a sphere.
scientific article; zbMATH DE number 4194624