On convex hypersurfaces in a space form
The aim of this paper is to generalize some known results which are valid for an ovaloid of \(E^ n\). The main result is the following: Let f: \(M\to \bar M\) be an isometric immersion of a smooth, compact, connected and orientable n-dimensional Riemannian manifold M into an \((n+1)\)- dimensional space form \(\bar M(c)\). If the sectional curvature K of M is nonnegative (when \(c<0)\) or \(K>c\) (when \(c\geq 0)\) then \[ \int_{M}\nabla_{II}(Tr L,\log Tr L)dM\geq 0\quad and\quad \int_{M}\nabla_{II}(Tr L^{-1},\det L)dM\leq 0 \] where L is the Weingarten map of f and \(\nabla_{II}\) is the first Beltrami operator with respect to the second fundamental form. Equality holds iff f is totally umbilical (when \(c\geq 0)\) or, either f is totally umbilical or L has two distinct constant eigenvalues at every point (when \(c<0)\).
scientific article; zbMATH DE number 4100117