On Conformal Killing Vector Fields on a Five-Dimensional 2-Symmetric Indecomposable Lorentzian Manifold with a Trivial Weyl Tensor

Abstract

The study of conformal transformation groups, Ricci flows, and Ricci solitons on various classes of manifolds is one of the actual problems of differential geometry. (Pseudo) Riemannian k-symmetric spaces are one of the important classes of such manifolds. If for the Riemannian case k=1, then there are k-symmetric spaces existing for any k values for the pseudo-Riemannian case. The generalized k-symmetric Kahen — Wallach spaces, as well as the 2- and 3-symmetric pseudo-Riemannian spaces, are good examples of that. They appear in studies of pseudo-Riemannian geometry and in physics, and have been studied by many mathematicians. These spaces and conformal Killing vector fields on them were studied by D.N. Oskorbin, E.D. Rodionov for the case of low dimensions. A connection between Ricci solitons and conformal Killing vector fields on the generalized k-symmetric Kahen — Wallach spaces was established. Also, it was found that the behavior of the conformal multiplier depends on the properties of the Weyl tensor. In this paper, new nontrivial examples of conformal Killing vector fields with a variable conformal factor on a fivedimensional 2-symmetric indecomposable Lorentzian manifold with zero Weyl tensor are constructed.