  New general results of non-existence and rigidity of spacelike submanifolds
immersed in a spacetime, whose mean curvature is a time-oriented causal vector
field, are given. These results hold for a wide class of spacetimes which
includes globally hyperbolic, stationary, conformally stationary and pp-wave
spacetimes, among others. Moreover, applications to the Cauchy problem in
General Relativity, are presented. Finally, in the case of hypersurfaces, we
also obtain significant consequences in Geometrical Analysis, solving new
Calabi-Bernstein and Dirichlet problems on a Riemannian manifold.
