Lightlike Hypersurfaces of Semi-Riemannian Manifolds

Abstract

Here we develop a theory on the differential geometry of a lightlike hypersurface M of a proper semi-Riemannian manifold $\bar M$ For this purpose, we introduce a non-degenerate screen distribution and construct the corresponding lightlike transversal vector bundle tr(TM) of M, consistent with the well-known theory of Riemannian submanifolds. This enables one to define the induced geometrical objects such as linear connection, second fundamental form, shape operator, etc., and to obtain the Gauss-Codazzi equations leading to the Fundamental Theorem of lightlike hyper-surfaces. It is noteworthy that the second fundamental form (and, therefore, the results on totally geodesic and totally umbilical lightlike hypersurfaces) is independent of the choice of a screen distribution.