Rigidity theorems for hypersurfaces in real space form (Q2781724)

Let \(M\) be a compact hypersurface in the space form \(R^{n+ 1}(c)\) of constant sectional curvature \(c\). Let \(B\) be the second fundamental form, \(H\) the mean curvature, \(\nabla B\) and \(\nabla H\) their covariant derivative. The author shows that if (i) \(|\nabla B|^2\geq |\nabla H|^2\) and \(|B|^2\leq 2c\sqrt{n-1}\), then either \(M\) is totally umbilical, or \(|B|^2= 2c\sqrt{n-1}\). The key point in the proof is an integral formula given by \textit{M. Okumura} [Am. J. Math. 96, 207-213 (1974; Zbl 0302.53028)].