Classification of Petrov Homogeneous Spaces

Abstract

In this paper the final stage of the Petrov classification is carried out. As it is known, the Killing vector fields specify infinitesimal transformations of the group of motions of space $V_4$. In the case when in the homogeneous space $V_4$ the group of motions $G_3$ acts simply transitive, the geometry of the non-isotropic hypersurface is determined by the geometry of the transitivity space $V_3$ of the group $G_3$. In this case, the metric tensor of the space $V_3$ can be given by a nonholonomic reper consisting of three independent vectors $\ell_{(a)}^\alpha$, which define the generators of the group $G_3$ of finite transformations in the space $V_3$. The representation of the metric tensor of $V_4$ spaces by means of vector fields $\ell_{(a)}^\alpha$ has a great physical meaning and allows to simplify substantially the equations of mathematical physics in such spaces. Therefore, the Petrov classification should be complemented by the classification of vector fields $\ell_{(a)}^\alpha$ connected to Killing vector fields. For homogeneous spaces this problem has been largely solved. A complete solution of this problem is presented in the present paper, where the Petrov classification for homogeneous spaces in which the group $G_3$, which belongs to type $VIII$ according to the Petrov classification, acts simply transitively, is refined. In addition, the complete classification of vector fields $\ell_{(a)}^\alpha$ for spaces $V_4$ in which the group $G_3$ acts simply transitivity on isotropic hypersurfaces.}