On the first Betti number of spacetimes with parallel lightlike vector field

Abstract

We prove that a non-totally vicious n-dimensional compact spacetime (M,g) admitting a parallel lightlike vector field is foliated by compact totally geodesic null hypersurfaces. As a consequence, assuming non-negative Ricci curvature on the leaves then the first Betti number of M is bounded above by n with equality if and only if M is diffeomorphic to the torus.