Submanifolds with Positive Ricci Curvature in the Constant Curvature Space

Let S~(n+p)(1) be a unit sphere,M~n an n-dimensional compact submanifold immersed in S~(n+p)(1) with non-zero parallel mean curvature vector.It is proved that when n≥4 and p≥2,if the Ricci curvature of M~n is not less than(n - 2)(1 + H~2),then M~n is totally umbilical,or the Ricci curvature of M~n equals to(n-2)(1+H~2),and so the geometry classification of M~n is given.