A characterization of round spheres
Sung-Eun Koh · Proceedings of the American Mathematical Society · 1998
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 n dimensional compact oriented manifold without boundary into n + 1 n+1 dimensional Euclidean space, hyperbolic space or the open half sphere is a totally umbilic immersion if, for some r , r = 2 , … , n , r, r=2,\dots ,n, the ( r − 1 ) (r-1) -th mean curvature H r − 1 H_{r-1} does not vanish and the ratio H r / H r − 1 H_{r}/H_{r-1} is constant.