Nonnegative Sectional Curvature Hypersurfaces in a Real Space Form

Shichang Shu ; Annie Yi Han

Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 776-793

Résumé

In this paper, we investigate the nonnegative sectional curvature hypersurfaces in a real space form $M^{n+1}(c)$. We obtain some rigidity results of nonnegative sectional curvature hypersurfaces $M^{n+1}(c)$ with constant mean curvature or with constant scalar curvature. In particular, we give a certain characterization of the Riemannian product $S^k(a)\times S^{n-k}(\sqrt{1-a^2})$, $1\le k\le n-1$, in $S^{n+1}(1)$ and the Riemannian product $H^k(\operatorname{tanh}^2r-1)\times S^{n-k}(\operatorname{coth}^2r-1)$, $1\le k\le n-1$, in $H^{n+1}(-1)$.

Détail

Mots-clés : hypersurface in Euclidean $n$-space
Keywords: space form, mean curvature, scalar curvature, principal curvature, sectional curvature, umbilical sphere, Codazzi equation, Ricci identity.