Lie group of the second degree infinitesimal conformal transformations in a symmetric Riemannian space of the first class

Abstract

The study of infinitesimal transformations in Riemannian spaces is of interest both theoretically and as an application. The distribution of relativistic gas according to the Maxwell-Boltzmann law is characterized by a vector ξi(x), which is a Killing vector (if the gas consists of particles of nonzero rest mass) or an infinitely small conformal transformation vector (if the gas consists of particles of zero rest mass). Groups of conformal transformations have been studied to a much lesser extent than movement groups. The group of conformal transformations in spherically symmetric gravitational fields was investigated by Takeno, who indicated the complete system of solutions of the generalized Killing equations for the indicated spaces.A.Z.Petrov gave a classification of gravitational fields of general form according to the groups of infinitesimal motions and conformal transformations. A complete solution to the problem of classifying gravitational fields by groups of conformal transformations was obtained by R.F.Bilyalov, the main result of which is as follows: the group of conformal transformations acting in a non-conformally flat gravitational field is a group of motions or homotheties of a space conformal to a given one.In this article, infinitesimal motions in symmetric Riemannian spaces of the first class Vn were studied. For n = 4 the basis of the Lie group G12 of examined transformations is explicitly found and the structure of this group is given.