On radius, systole, and positive Ricci curvature
Recall that the radius of a compact metric space, \((X, \text{dist})\) is defined by \[ \text{Radius } X = \min_{x \in X} \max_{y \in X} \text{dist} (x,y). \] In this note we prove that a closed Riemannian \(n\)- manifold with \(\text{Ricci }M \geq n -1\) is diffeomorphic and almost isometric to \(S^ n\), if its radius is sufficiently close to \(\pi\) compared to its lower bound for sectional curvature. We also show that the same conclusion can be obtained if we replace the hypothesis on diameter, provided also that the \(d\)-excess of \(M\) is sufficiently small. As a corollary of the radius theorem we get that \(M\) is diffeomorphic and almost isometric to \(\mathbb{R} P^ n\), if \(\text{Ricci }M \geq n - 1\) and \(\text{Sys}_ 1M\) is sufficiently close to \(\pi\) compared to the minimum of the sectional curvatures of \(M\). Here \(\text{Sys}_ 1 M\) denotes the length of the shortest closed noncontractible curve in \(M\).
scientific article; zbMATH DE number 784092
scientific article