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 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
dimM=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.