Spacelike submanifolds in the de Sitter space S p n+p (c) with constant scalar curvature

Abstract

Let M n be a closed spacelike submanifold isometrically immersed in de Sitter space S p n+p (c). 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 M n, respectively. Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for M n immersed in S p n+p (c) with parallel normalized mean curvature vector field is proved. When n≥3, the pinching constant is the best. Thus, the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature” (see Manus Math, 1998, 95:499\s-505) is corrected. Moreover, the reduction of the codimension when M n is a complete submanifold in S p n+p (c) with parallel normalized mean curvature vector field is investigated.