On space–time carrying a total hyperbolic skew symmetric Killing vector field

Abstract

The complex vectorial formalism on a general space–time ( M, g) was constructed by Cahen, Debever and Defrise. This formalism is based on the local isomorphism I: L(4)→SO 3( C) , where L(4) is the four-dimensional Lorentz group acting on the tangent spaces T p M and SO 3( C) is the three-dimensional complex rotation group. In this framework, the congruence of Debever plays a distinguished role. Its properties determine the general space–time M, in terms of Petrov’s classification. In the present paper, we assume that any hyperbolic vector field X on M is a skew symmetric Killing vector field having a spatial vector field Y as generative. The existence of such a vector field X is determined by an exterior differential system in involution. It is shown that M is the local Riemannian product M= M h× M s, where M h (resp. M s) is a totally geodesic and totally pseudo-isotropic hyperbolic (resp. spatial) surface (the Gauss map is ametric). Any such M is a space–time of type D in Petrov’s classification. It is proved that the congruence of Debever is of electric type; in particular, it is geodesic and shear 1-free. Other geometric properties on such a general space–time are obtained.