On spherical submanifolds with finite type spherical Gauss map

Chen and Lue (2007) initiated the study of spherical submanifolds with finite type spherical Gauss map. In this paper, we firstly prove that a submanifold M-n of the unit sphere Sm-1 has non-mass-symmetric 1-type spherical Gauss map if and only if M-n is an open part of a small n-sphere of a totally geodesic (n+1)-sphere Sn+1 subset of Sm-1. Then we show that a non-totally umbilical hypersurface M of Sn+1 with nonzero constant mean curvature in Sn+1 has mass-symmetric 2-type spherical Gauss map if and only if the scalar curvature curvature of M is constant. Finally, we classify constant mean curvature surfaces in S-3 with mass-symmetric 2-type spherical Gauss map.