Notes on Hypersurfaces in a Riemannian Manifold

Source abstract

H. Liebmann (3) and W. Süss (7) proved Theorem A. The only convex closed hypersurface with constant mean curvature in a Euclidean space is a sphere . Y. Katsurada (1; 2) gave the following generalization. Theorem B. Let M be an orientable Einstein space which admits a proper conformai Killing vector field, that is, a vector field generating a local one-parameter group of conformai transformations which is not that of isometries, and S a closed orientable hypersurface in M whose first mean curvature is constant. If the inner product of the conformai Killing vector field and the normal to the hypersurface has fixed sign on S, then every point of S is umbilical .