  Some analysis on the Lorentzian distance in a spacetime with controlled
sectional (or Ricci) curvatures is done. In particular, we focus on the study
of the restriction of such distance to a spacelike hypersurface satisfying the
Omori-Yau maximum principle. As a consequence, and under appropriate hypotheses
on the (sectional or Ricci) curvatures of the ambient spacetime, we obtain
sharp estimates for the mean curvature of those hypersurfaces. Moreover, we
also give a suficient condition for its hyperbolicity.
