Complete spacelike hypersurfaces on symmetric spacetimes

A Lorentz manifold is said to be a conformally stationary spacetime if it is endowed with a globally defined conformal timelike vector field K, whereas it is a pp-wave when there is a globally defined parallel lightlike vector field K on M. The study of rigidity and non-existence results for spacelike hypersurfaces with constant mean curvature in these spaces has been considered in the last years by several authors. In this note we unify some known techniques and results related to both problems by considering spacelike hypersurfaces in a spacetime endowed with a globally defined conformal causal vector field. This wide family of Lorentz manifolds not only includes previous cases, but it also includes new families of spacetimes. (note)