A rigidity theorem for submanifolds in S^n＋p with constant scalar curvature

Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn+p. Denote by R, H and S, the normalized scalar curvature, the mean curvature, and the square of the length of the second fundamental form of Mn, respectively. Suppose R is constant and ≥1. We study the pinching problem on S and prove a rigidity theorem for Mn immersed in Sn+p with parallel normalized mean curvature vector field. When n≥8 or, n=7 andp≤2, the pinching constant is best.