Bulletin of the
Korean Mathematical Society BKMS

A characterization of geodesic spheres in the simply connected space forms in terms of the ratio of the Gauss-Kronecker curvature and the (usual) mean curvature is given: An immersion of $n$ dimensional compact oriented manifold without boundary into the $n+1$ dimensional Euclidean space, hyperbolic space or open half sphere is a totally umbilic immersion if the mean curvature $H_1$ does not vanish and the ratio $H_n/H_1$ of the Gauss-Kronecker curvature $H_n$ and $H_1$ is constant.

Keywords: Gauss-Kronecker curvature, mean curvature, principal curvature, umbilical point, Minkowski formula

MSC numbers: Primary 53C40, 53C42