On the squares of umbilicity factors of null hypersurfaces in indefinite Kaehler manifolds

Abstract

Any null hypersurface (M,g) of an indefinite Kaehler manifold, denoted as (\overline{M},\overline{g},\overline{J}), is endowed with two special null vector fields U:=-\overline{J}N and V:=-\overline{J}\xi, where N and \xi, respectively, span the transversal and normal bundle (radical distribution) to M. In this paper, we define the U and V-null sectional curvatures, and then apply them to totally umbilic null hypersurfaces. In particular, we prove that the squares of the umbilicity factors measures, up to third order, the differences in lengths of any two spacelike geodesics tangent to the \overline{J}-invariant distribution D_{0} over M.