Conformally Flat Ricci Solitons on Lie Groups with Left-invariant (pseudo)Riemannian Metrics
Abstract
Ricci solitons are important generalizations of Einstein metrics on pseudo-Riemannian manifolds, which were first considered by R. Hamilton. Homogeneous Ricci solitons have been studied by many mathematicians, and the classification of homogeneous Ricci solitons is known in small dimensions only and it is not exhaustive. Algebraic Ricci solitons are important subclass of the homogeneous Ricci solitons, and algebraic Ricci solitons were first discussed by J. Lauret in connection with the study of homogeneous Ricci solitons on solvable Lie groups. The study of algebraic Ricci solitons is important, since it is known that every algebraic Ricci soliton is homogeneous Ricci soliton. In this paper, we study the algebraic Ricci solitons on Lie groups with left-invariant pseudo- Riemannian metric, if metric is conformally flat and the Ricci operator are diagonalizable. In this case it was possible to show that the Ricci soliton is trivial, t.i. metric Lie group is Einstein manifold or the direct product of Einstein manifold and the Euclidean space.We also obtained a consequence of the absence of nontrivial homogeneous invariant Ricci solitons on Lie groups with left-invariant conformally flat Riemannian metric.
DOI 10.14258/izvasu(2016)1-22
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