Higher Mean Curvature And Rigidity Theorems Of Spheres

We set up a new characterization of geodesic hyperspheres in the space forms and generalize Koh’s concerned result. Furthermore, we consider the totally um-bilical properties of complete submanifolds with parallel mean curvature vector in spheres, and generalize a result of Alencar, do Carmo and Santos to the complete submanifolds. The results of this paper are as follows.Theorem1Let Mn be a compact oriented n-dimensional Riemannian manifold without boundary, and Nn+1be one of the Euclidean space Rn+1, the hyperbolic space Hn+1or the open half sphere S+n+1. Let φ:Mn→Nn+1be an isometric immersion. We denote the r-th mean curvature of Mn by Hr. If there is a point of Mn where all the principal curvatures are positive, for some integers s and r, s<r, s, r∈{1,..., n}, Hr>0, Hs/Hr=a, where a is a constant, then Mn is a geodesic hypersphere.Corollary Let Mn be a compact oriented embedded hypersurface in Nn+1. If Hr≠0, Hs/Hr is a constant, s<r, s,r∈{0,...,n}, then Mn is a geodesic hypersphere.Theorem2Let Mn be a complete submanifold with parallel mean curva- ture vector in the unit sphere Sn+P. Suppose that and AP,H is the square of the positive roots of If sup|S-nH2|2<Ap,H, where H is the mean curvature and S is the square length of the second foundamental form of Mn, then Mn is totally umbilical.