On Homogeneous Ricci Solitons on Three-Dimensional Locally Homogeneous (Pseudo)Riemannian Spaces with a Semisymmetric Connection
УДК 514.765
Abstract
Ricci solitons are a natural generalization of Einstein metrics and represent a solution to the Ricci flow. In the general case, they were studied by many mathematicians, which was reflected in the reviews by H.-D. cao, R.M. Aroyo — R. Lafuente. This issue has been most studied in the homogeneous Riemannian case, as well as in the case of trivial Ricci solitons, or Einstein metrics. In this paper, we study homogeneous Ricci solitons on three-dimensional locally homogeneous (pseudo) Riemannian spaces with a nontrivial isotropy group and a semisymmetric connection. A classification of homogeneous Ricci solitons on three-dimensional locally homogeneous (pseudo) Riemannian spaces with a semisymmetric connection is obtained. It is proved that in the case of Lie groups there exist nontrivial invariant Ricci solitons. Earlier, L. cerbo showed that all invariant Ricci solitons are trivial or Einstein metrics on unimodular Lie groups with a left-invariant Riemannian metric and a Levi-Civita connection. In the non-unimodular case, a similar result was obtained by P.N. Klepikov and D.N. Oskorbin up to dimension four inclusively. The problem remains open for the cases of dimension 5 and higher.
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Copyright (c) 2024 Виталий Владимирович Балащенко, Павел Николаевич Клепиков, Евгений Дмитриевич Родионов, Олеся Павловна Хромова
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