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⊂(M,g) of general type and for some special lightlike hypersurfaces (namely, for totally geodesic, umbilical, and belonging to a manifold (M,g) of constant curvature), in a third-order neighborhood of a point x∈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.