Ricci Solitons on 2-Symmetric Four-Dimensional Lorentzian Manifolds
Abstract
An important generalization of Einstein metrics on a (pseudo) Riemannian manifolds are Ricci solitons first discussed by R. Hamilton. The problem of finding Ricci solitons is quite difficult, so we assume the restriction of one of the following: the structure of the manifold, the dimension, the class of metrics, or a class of vector fields, participating in the Ricci soliton equation. The most important examples of such restrictions are 2-symmetric Lorentzian manifolds investigated by A.S. Galaaev, D.V. Alekseevskii, and J.M. Senovilla. 2-Symmetric locally indecomposable Lorentzian manifolds have parallel null-distribution, i.e. they are Walker manifolds. These manifolds have a special coordinate system, which allows us to solve Ricci soliton equation locally. In this paper, we investigate the Ricci soliton equation on 2-symmetric locally indecomposable Lorentzian manifolds. K. Onda and B. Batat investigated Ricci solitons on the four-dimensional 2-symmetric Lorentzian manifolds. Local solvability of the Ricci soliton equation on such manifolds was proven. In this paper, we obtain general solution of the Ricci soliton equation on four-dimensional 2-symmetric locally indecomposable Lorentzian manifolds.
DOI 10.14258/izvasu(2017)4-23
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References
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Copyright (c) 2017 Д. Н. Оскорбин, Е. Д. Родионов, И.В. Эрнст
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