Abstract
We study complete noncompact spacelike hypersurfaces immersed into conformally stationary spacetimes, equipped with either one or two conformal vector fields. In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of totally geodesic hypersurfaces in terms of their $r$-th mean curvatures. For instance, for a timelike geodesically complete conformally stationary spacetime endowed with a closed conformal timelike vector field $V$, under appropriate restrictions on the flow and the norm of the tangential component of $V$, we are able to prove that totally geodesic spacelike hypersurfaces must be, in fact, leaves of the distribution determined by $V$. Applications to the so-called generalized Robertson--Walker spacetimes are also given. Furthermore, we extend our approach in order to obtain a lower estimate of the relative nullity index.
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