Submanifilds With Parallel M(?)BIUS Form In S～n

Let x : M→ S~n be a submanifold in the unit sphere S~n without umbilic points. Four basic invariants of M under the mobius transformation group of S~n are: mobius metric g, mobius form φ, mobius second form B and Blaschke tensor A. Firstly, the submanifolds with parallel mobius form and constant mobius scalar curvature are studied and the respectively rigidity theorems are given. Secondly, it is proved that a surface with parallel mobius form must be a surface with vanishing mobius form. So the classification of the surface with parallel mobius form was given. Finally,the characterization of submanifolds in real space forms with parallel mean curvature vector fields and constant scalar curvature are shown.