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Journal of Lie Theory 29 (2019), No. 4, 957--968
Copyright Heldermann Verlag 2019

Left Invariant Ricci Solitons on Three-Dimensional Lie Groups

Hamid R. Salimi Moghaddam
Department of Mathematics, Faculty of Sciences, University of Isfahan, Isfahan 81746-73441, Iran

We give a necessary and sufficient condition for an arbitrary real Lie group, to admit an algebraic Ricci soliton. As an application, we classify all algebraic Ricci solitons on three-dimensional real Lie groups, up to automorphism. This classification shows that, in dimension three, there exist a solvable Lie group and a simple Lie group such that they do not admit any algebraic Ricci soliton. Also it is shown that there exist three-dimensional unimodular and non-unimodular Lie groups with left invariant Ricci solitons. Finally, for a unimodular solvable Lie group, the solution of the Ricci soliton equation is given, explicitly.

Keywords: Ricci soliton, left invariant Riemannian metric, three-dimensional Lie group.

MSC: 22E60, 53C44, 53C21.

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