On the scalar curvature of spacelike hypersurfaces in generalized Robertson Walker spacetimes

Abstract

In this work we study the scalar curvature S of a spacelike hypersurface M immersed in a Generalized Robertson Walker (GRW) spacetime M‾=I×fF. Specifically, we express S in terms of the Ricci curvature of the fiber F, the warping function f and the shape operator of the immersion. As a consequence, under natural hypothesis we provide several characterizations of the spacelike slices, as well as the totally umbilical spacelike hypersurfaces, of a GRW spacetime. Moreover, we particularize our study to important GRW spacetimes as Einstein GRW spacetimes, static GRW spacetimes, the Einstein de Sitter spacetime, the steady state spacetime, and spatially parabolic GRW spacetimes. Moreover, we prove that there is no complete totally geodesic spacelike hypersurfaces in a family of Generalized Robertson Walker spacetimes which includes the steady state spacetime.