A characterization of round spheres

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 $n$ dimensional compact oriented manifold without boundary into $n+1$ dimensional Euclidean space, hyperbolic space or the open half sphere is a totally umbilic immersion if, for some $r, r=2,\dots ,n,$ the $(r-1)$-th mean curvature $H_{r-1}$ does not vanish and the ratio $H_{r}/H_{r-1}$ is constant.