On the Ricci Curvature of Compact Spacelike Hypersurfaces in Einstein Conformally Stationary-Closed Spacetimes

Abstract

In this paper we develop an integral formula involving the Ricci and scalar curvatures of a compact spacelike hypersurface M in a spacetime \(\overline M \) equipped with a timelike closed conformal vector field K (in short, conformally stationary-closed spacetime), and we apply it, when \(\overline M \) is Einstein, in order to establish sufficient conditions for M to be a leaf of the foliation determined by K and to obtain some non-existence results. We also get some interesting consequences for the particular case when \(\overline M \) is a generalized Robertson-Walker spacetime.