Abstract
Let M n be a closed spacelike submanifold isometrically immersed in de Sitter space S n+pp (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 n+pp (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 n+pp (c) with parallel normalized mean curvature vector field is investigated.
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Jianfeng, Z. Spacelike submanifolds in the de Sitter space S n+pp (c) with constant scalar curvature. Appl. Math. Chin. Univ. 20, 183–196 (2005). https://doi.org/10.1007/s11766-005-0051-3
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DOI: https://doi.org/10.1007/s11766-005-0051-3