Ricci curvature and the topology of open manifolds
Let \((M, g)\) be an \(n\)-dimensional complete Riemannian manifold. The authors consider a global metric invariant \(A\) which expresses the \(r\)-limiting value of the ratio \(\text{Vol}(B(p,r))/E(n,r)\), where \(B(p,r)\) denotes the geodesic ball with radius \(r\), centered in \(p\in M\); \(E(n,r)\) represents the volume of the Euclidean \(n\)-ball with radius \(r\). The space \((M, g)\) has large volume growth if \(A\) is positive. In this paper, the analogous of the theorem of \textit{V. B. Marenich} and \textit{V. T. Toponogov} [Sib. Mat. Zh. 26, 191-194 (1985; Zbl 0578.53030)] for the case of spaces with large volume growth and with nonnegative Ricci curvature is obtained: the theorem of Marenich-Toponogov states that an \(n\)-dimensional complete manifold with nonnegative sectional curvature and large volume growth is diffeomorphic to a Euclidean \(n\)-space.
scientific article; zbMATH DE number 1423137
scientific article