Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in De Sitter space

On spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

“…An interesting result of Cheng and Ishikawa [6] states that the totally umbilical round spheres are the only compact spacelike hypersurfaces in S n+1 1 (1) with constant normalized scalar curvature R < 1. Some other authors, such as Brasil, Colares and Palmas [3], Camargo, Chaves and Sousa Jr. [4], Caminha [5], Hu, Scherfner and Zhai [10] and Li [11] have also worked on related problems.…”

“…So Theorem 1.2(i) can be viewed as a kind of extension of the result due to F.E.C. Camargo et al in [4], saying that a complete spacelike hypersurface M n (n 3) in the de Sitter space S n+1 1 (c) with constant normalized scalar curvature R satisfying n−2 n c R c must be totally umbilical provided that M n has bounded mean curvature. On the other hand, consider the spacelike hypersurface embedded into S n+1…”

Linear Weingarten spacelike submanifolds in de Sitter space

“…Caminha [7] answered this question under the additional condition that the supremum of H is attained on M n . Later, Camargo-Chaves-Sousa Jr [6] answered the question affirmatively under the condition that H is bounded. They also proved that complete spacelike hypersurfaces with constant R < 1 are totally umbilical if sup S < 2 √ n − 1.…”

Complete spacelike hypersurfaces in conformally stationary Lorentz manifolds

“…For the de Sitter space, Brasil Jr., Colares and Palmas also used the Omori-Yau maximum principle in [6] to characterize the hyperbolic cylinders as the only complete hypersurfaces in the de Sitter space with constant mean curvature, nonnegative Ricci curvature and having at least two ends (see also [7] for the case of the scalar curvature).…”