Ricci curvature of spacelike hypersurfaces in de Sitter space

In this paper, we study the spacelike hypersurfaces in de Sitter space S1n+1(c), and give some estimates on the Ricci curvature tensor and the square of its length. By these estimates, many properties of spacelike hypersurfaces are derived, that is, the compactness and convexity of spacelike hypersurfaces, the finiteness of fundamental group. We also get an integral inequality on eigenvalue of the Laplacian operator and a sufficient and necessary condition for such hypersurfaces to be totally geodesic.