Spacelike hypersurfaces in twisted product spacetimes with complete fiber and Calabi-Bernstein-type problems

In this article spacelike hypersurfaces immersed in twisted product spacetimes $I\times_f F$ with complete fiber are studied. Several conditions ensuring global hyperbolicity are presented, as well as a relation that needs to hold on each spacelike hypersurface in $I\times_f F$ for it to be a simple warped product. When the fiber is assumed to be closed (compact and without boundary) and the ambient spacetime has a suitable expanding behaviour, non-existence results for constant mean curvature hypersurfaces are obtained. Under the same hypothesis, a characterization of compact maximal hypersurfaces and other for totally umbilic ones with a suitable restriction on their mean curvature are presented. The description of maximal hypersurfaces in twisted product spacetimes of the form $I\,{ }_{f}\!\!\times F$ with a one-dimensional Lorentzian fiber is also included. Finally, the mean curvature equation for a spacelike graph on the fiber is computed and as an application, some Calabi-Bernstein-type results are proven. We also include in an Appendix some known conformal geometry results describing the transformation of relevant tensors and operators under the action of a conformal map in a pseudo-Riemannian background.