P.N. Klepikov, D.N. Oskorbin
Homogeneous Invariant Ricci Solitons on Four-dimensional Lie Groups
Ricci solitons are important generalizations of Einstein metrics on Riemann manifolds. These metrics were first investigated by Hamilton. Ricci solitons are relevant to the solutions of the Ricci flow. Homogeneous Riemannian metric on the homogeneous space G/H satisfying the Ricci soliton is called the homogeneous Ricci soliton. Such metrics have been studied by many mathematicians. The classification of homogeneous Ricci solitons is known in small dimensions only, and it is not exhaustive. It is known that for three-dimensional Lie groups with left-invariant Riemannian metric Ricci soliton equation has no solution in the class of left-invariant vector fields. A similar fact is proved for unimodular Lie groups with left-invariant Riemannian metric of any finite dimension. However, the existence problem for non-trivial invariant Ricci solitons on nonunimodular Lie groups of dimension > 3 remains open. In this paper, we obtain the solution of this problem in dimension 4. The soliton equation by generalized Milnor’s frames reduced to the system of polynomial equations. The absence of nontrivial homogeneous invariant Ricci solitons on fourdimensional Lie groups is proved.
Key words: Lie group, Lie algebra, invariant Ricci soliton, left-invariant Riemannian metric, J. Milnor’s generalized bases
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