Department of Mathematics, Kon-Kuk University, Seoul, 143-701, Korea
Abstract:
A new characterization of geodesic spheres in the simply connected space forms in terms of higher order mean curvatures is given: An immersion of an dimensional compact oriented manifold without boundary into dimensional Euclidean space, hyperbolic space or the open half sphere is a totally umbilic immersion if, for some the -th mean curvature does not vanish and the ratio is constant.