A rigidity phenomenon on Riemannian manifolds with reverse volume pinching
The well-known topological sphere theorem states that a complete, simply-connected, n-dimensional Riemannian manifold with sectional curvature taking values in the interval \((1,4]\) is homeomorphic to the \(n\)-sphere. The proof of this celebrated theorem undergone several steps and was computed in 1961 by Klingenberg. The differentiable sphere theorem which claims that the manifold is actually diffeomorphic to the \(n\)-sphere with the standard differentiable structure under the same hypothesis was recently proved by [\textit{S. Brendle} and \textit{R. Schoen}, J. Am. Math. Soc. 22, 287--307 (2009; doi:10.1090/S0894-0347-08-00613-9)]. The main result of the paper under review is that a compact simply connected \(2n\)-dimensional Riemannian manifold without boundary is diffeomorphic to the sphere \(S^{2n}\) if its curvature and volume are bounded by: \(0<K_M \leq 1\), \(V(M)\leq (1+\eta)V(S^{2n})\), where \(\eta\) is a constant dependent only on \(n\). The topological version of this was proved in [\textit{D. Coghlan} and \textit{Y. Itokawa}, Proc. Am. Math. Soc. 111, No. 3, 815--819, (1991; Zbl 0719.53019)] where the same curvature bound was used whilst the volume bound was \(V(M)< \frac{3}{2}V(S^{2n})\).
scientific article; zbMATH DE number 5306111
scientific article