On the totally geodesic spacelike hypersurfaces in conformally stationary spacetimes

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.