In this talk, we consider spacelike hypersurfaces with proper second mean curvature vector field of four dimensional de Sitter space S^4_1. First, we show that such a hypersurface has constant scalar curvature. Also, we show that each spacelike hypersurface of S^4_1 with proper second mean curvature vector field is Box-biharmonic, $\Box$-1-type or Box-null-2-type. Finally, we prove that every spacelike hypersurface with proper second mean curvature and constant ordinary mean curvature in de Sitter 4-space is 1-maximal.
