О киллинговых полях на 2-симметрических лоренцевых многообразиях

Abstract

In the following paper, we describe Killing fields on 2-symmetric Lorentzian manifolds of dimension four. Killing fields play an important role in the study of Ricci solitons which were introduced by R. Hamilton. Ricci solitons are the generalization of the Einstein metrics on (pseudo)Riemannian manifolds. Ricci soliton equation was studied by many mathematicians on different classes of manifolds. In particular, in the recent author’s papers, solvability of the Ricci soliton equation on 3-symmetric Lorentzian manifolds was proved, and the general solution of the Ricci soliton equation on 2-symmetric Lorentzian manifolds was described. We describe Killing fields using Brinkmann normal coordinates which exist on the class of Lorentzian manifolds, the so-called pp-waves. The system of differential equations that corresponds to the Killing equation can be reduced to a simpler form. This was done by W. Globke and T. Leistner. By applying their result, the general solution of the system was found, the dimension of an algebra of Killing fields was calculated. The results stated in this paper continue author’s research on Ricci solitons on Lorentzian manifolds.