Semi Riemannian hypersurfaces with a canonical principal direction

Abstract

We study semi-Riemannian hypersurfaces with a canonical principal direction (CPD) with respect to a nondegenerate closed conformal vector field on a semi-Riemannian ambient manifold. We give a characterization of such hypersurfaces. In the case when such hypersurface is a surface with constant mean curvature in a semi-Riemannian space form, we prove that it has an intrinsic Killing vector field. A special case of hypersurfaces with a CPD are those with constant angle with respect to a parallel vector field in the semi-Riemannian ambient. We prove that a surface with zero mean curvature and constant angle, in a Loretzian ambient of arbitrary dimension, is necessarily flat. When the surface is timelike and the ambient has non positive curvature then the surface is totally geodesic. When the surface is spacelike and the ambient has non negative curvature then the surface is totally geodesic. In general when the ambient is of dimension three then the surface is always totally geodesic.