A rigidity theorem for compact hypersurfaces with an upper bound for the Ricci curvature (Q1282308)

There are classical results on the diameter of a complete Riemannian manifold with Ricci curvature bounded below (Bonnet-Myers theorem, Cheng maximal diameter sphere theorem). Here, a result for a compact hypersurface \(N\) of a space form \(\widetilde S_{n+1}(c)\) \((n\geq 3)\) with Ricci curvature Ric of \(N\) bounded above is presented. Let \(N\) be included in a closed normal ball \(\overline B_p(r)\) with \(r<\pi/(2\sqrt c)\) if \(c>0\) and assume that \(\text{Ric}\leq (n-1)(c+ k^2_c(r))\). Then, \(N= \partial\overline B_p(r)= S_p(r)\). This result is obtained as a corollary of a more general theorem, where, instead of \(\widetilde S_{n+1}(c)\), a complete Riemannian manifold \(M\) with sectional curvature \(K^M\) bounded from above by a constant \(c\) is considered, and the Ricci curvature of \(N\) satisfies a more general inequality.