Smooth diameter and eigenvalue rigidity in positive Ricci curvature (Q2750918)

Sphere theorems typically state that a Riemannian manifold satisfying certain curvature bounds and restrictions on other geometric quantities (diameter, volume) are homeomorphic, or even diffeomorphic, to the standard sphere. In this article, the author combines previous sphere theorems, in particular a topological sphere theorem by \textit{G. Perelman} [Math.~Z. 218, 595-596 (1995; Zbl 0831.53033)], with a recent estimate for the injectivity radius in [\textit{A. Petrunin} and \textit{W. Tuschmann}, Geom. Funct. Anal. 9, 736-774 (1999; Zbl 0941.53026)] to prove the following smooth sphere theorem: any \(m\)-dimensional complete Riemannian manifold with Ricci curvature \(\text{Ric} \geq m-1\), sectional curvature~\(K\) bounded above and diameter sufficiently close to~\(\pi\), is diffeomorphic to the standard unit \(m\)-sphere. The same conclusion holds if, instead of the condition on the diameter, one requires the first eigenvalue of the Laplacian to be sufficiently close to~\(m\).