Null hypersurface normalized by the curvature vector field in a Lorentzian manifold of quasi-constant curvature

Abstract

We introduce a class of null hypersurfaces $M$ of Lorentzian manifolds of quasi-constant curvature $\overline{M}$, namely, $\zeta$-rigging null hypersurfaces, whose curvature vector field $\zeta$ is a rigging for $M$. We  extend some well-known results for Lorentzian manifolds of constant curvature and prove several classification theorems  for such a null hypersurface. Next, we establish sufficient conditions to guarantee that such a null hypersurface must be totally geodesic. As a consequence, we prove that the ambient manifold  is flat along $M$.