A rigidity theorem for submanifolds inSn+p with constant scalar curvature

LetM1 be a closed submanifold isometrically immersed in a unit sphereSn+p. Denote byR, H andS, the normalized scalar curvature, the mean curvature, and the square of the length of the second fundamental form ofM1, respectively. SupposeR is constant and ≥1. We study the pinching problem onS and prove a rigidity theorem forM1 immersed inSn+p with parallel normalized mean curvature vector field. Whenn≥8 or,n=7 andp≤2, the pinching constant is best.