Abstract: The compact n dimensional space-like hypersurfaces with constant scalar curvature in locally symmetric de Sitter space Nⁿ⁺¹₁ are studied, with some rigidity characterization theorems obtained by introducing a self-adjoint operator and by using the moving frame method. As a corollary, the rigidity characterization theorem is obtained on the space-like hypersurfaces in a de Sitter space Sⁿ⁺¹₁. These results generalize the results of Zheng and Liu, and the pinching constants do not depend on the dimensional n.

Key words: locally symmetric, de Sitter space, space-like hypersurfaces, totally umbilical, hyperbolic cylinder