Ricci Solitons on 2-symmetric Lorentzian Manifolds

Abstract

Ricci solitons are an important generalization of Einstein metrics on (pseudo) Riemannian manifolds, and this notion was introduced by R.Hamilton. The problem of solving the Ricci soliton equation is quite difficult, therefore one can assume some restrictions either on a structure of the manifold or on the dimension or on a class of metrics, or on a class of vector fields, which are contained in the Ricci soliton equation. Walker manifolds are one of the most important examples of such restrictions, that is pseudo-Riemannian manifolds admitting a smooth parallel (in sense of Levi-Civita connection) isotropic distribution. The geometry of Walker manifolds and Ricci solitons on them were studied by many mathematicians. In this paper, we investigate the Ricci soliton equation on some Lorentzian manifolds. In particular, we study the Ricci solitons on 2-symmetric Lorentzian manifolds, which are Walker manifolds, as it was proven by D.V. Alekseevsky and A.S. Galaev. K. Onda and B. Batat investigated Ricci solitons on fourdimensional 2-symmetric Lorentzian manifolds, and proved local solvability of the Ricci soliton equation on such manifolds. In this paper we have obtained local solvability of the Ricci soliton equation on fivedimensional 2-symmetric Lorentzian manifolds.