Lightlike hypersurfaces

Abstract

Since for any semi-Riemannian manifold $\bar M$ there is a natural existence of null (lightlike) subspaces, their study is equally desirable. In particular, from the point of physics lightlike hypersurfaces are of importance as they are models of various types of horizons, such as Killing, dynamical and conformal horizons, studied in general relativity (see some details in Chapter 3). However, due to the degenerate metric of a lightlike submanifold M, one fails to use, in the usual way, the theory of non-degenerate geometry. The primary difference between the lightlike submanifolds and the non-degenerate submanifolds is that in the first case the normal vector bundle intersects the tangent bundle. In other words, a vector of a tangent space $T_x \bar M$ cannot be decomposed uniquely into a component tangent to T _x M and a component of normal space T _x M^⊥. Therefore, the standard definition of the second fundamental form and the Gauss-Wiengarten formulas do not work, in the usual way, for the lightlike case.