We study the geometry of complete spacelike hypersurfaces with constant scalar curvature immersed into the de Sitter space. In this setting, we show that such a hypersurface must be totally umbilical, provided that its Gauss map has some suitable behavior. Our approach is based on the use of an appropriated generalized maximum principle, which can be seen as a sort of extension to complete Riemannian manifolds of the classical Hopf's maximum principle.
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