Ricci Solitons on 3-Symmetric Lorentzian Manifolds
An important generalization of Einstein equations on (pseudo)Riemannian manifolds is the Ricci soliton equation which was first discussed by R. Hamilton. The solving of the Ricci soliton equation becomes possible when there are some restrictions on the structure of the manifold, or the dimension, or the class of metrics, or a class of vector fields, which appears in the Ricci soliton equation. If there is a special coordinate system, then the problem of solving the Ricci soliton equation reduces to solving a system of PDE’s. There are Brinkman coordinates on Lorentzian Walker manifolds, which are Lorentzian manifolds with a parallel (in terms of Levi-Civita) distribution of isotropic lines. This fact allows one to investigate the Ricci soliton equation on these manifolds. 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 3-symmetric indecomposable Lorentzian manifolds. These manifolds have been studied by
D.V. Alekseevskii and A.S. Galaev, who have built a special local coordinate system. This article continues the authors’ study and the study of K. Honda and B. Batat, who have investigated Ricci solitons on 2-symmetric Lorentzian manifolds.
Hamilton R.S. Three manifolds with positive Ricci curvature // J. Diff. Geom. — 1982. — V. 17.
Cao H.-D. Recent progress on Ricci solitons // Advanced Lectures in Mathematics. — 2010. -V. 11.
Cerbo L.F. Generic properties of homogeneous Ricci solitons // Adv. Geom. — 2014. — V. 14.
Lauret J. Ricci soliton solvmanifolds // Journal fur die Reine und Angewandte
Mathematik. — 2011. — V. 650.
Walker A.G. On parallel fields of partially null vector spaces // Quart. J. Math., Oxford Ser. — 1949. — V. 20.
M. Brozos-Vazquez, E. Garcia-Rio, P.Gilkey, S.Nikcevic and R.Vazquez-Lorenzo. The geometry of Walker manifolds. Synthesis Lectures on Mathematics and Statistics // Morgan & Claypool Publ. — 2009.
Оскорбин Д.Н., Родионов Е.Д., Эрнст И.В. О солитонах Риччи на 2-симметрических четырехмерных лоренцевых многообразиях // Изв. Алт. гос. ун-та, 2017. — № 4.
Galaev A.S. Classification of third-order symmetric Lorentzian manifolds // Classical Quantum Gravity. — 2015. — V. 32, No. 2.
Globke W., Leistner T. Locally homogeneous pp-waves // Journal of Geometry and Physics. — 2016. — V. 108.
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