On some methods of construction of invariant normalizations of lightlike hypersurfaces

Abstract

The authors study the geometry of lightlike hypersurfaces on pseudo-Riemannian manifolds $(M, g)$ of Lorentzian signature. Such hypersurfaces are of interest in general relativity since they can be models of different types of physical horizons. For a lightlike hypersurface $V \subset (M, g)$ of general type and for some special lightlike hypersurfaces (namely, for totally umbilical and belonging to a manifold $(M, g)$ of constant curvature), in a third-order neighborhood of a point $x \in V$, the authors construct invariant normalizations intrinsically connected with the geometry of $V$ and investigate affine connections induced by these normalizations. For this construction, they used relative and absolute invariants defined by the first and second fundamental forms of $V$. The authors show that if $\dim M = 4$, their methods allow to construct three invariant normalizations and affine connections intrinsically connected with the geometry of $V$. Such a construction is given in the present paper for the first time. The authors also consider the fibration of isotropic geodesics of $V$ and investigate their singular points and singular submanifolds.