On an Equation in the Ricci Solitons Theory with a Semisymmetric Connection

Abstract

The study of Ricci solitons and invariant Ricci solitons with connections of various types has garnered much attention from many mathematicians. Metric connections with vector torsion, or semisymmetric connections, were first studied by E. Cartan on (pseudo) Riemannian manifolds. Later, K. Yano and I. Agricola studied tensor fields and geodesic lines of such connections, while P.N. Klepikov, E.D. Rodionov, and O.P. Khromova considered the Einstein equation of semisymmetric connections on three-dimensional locally homogeneous (pseudo) Riemannian manifolds. In the previous paper, the authors studied invariant Ricci solitons with a semisymmetric connection. They are an important subclass of the class of homogeneous Ricci solitons. We obtained the classification of invariant Ricci solitons on three-dimensional Lie groups with a left-invariant Riemannian metric and a semisymmetric connection different from the Levi-Civita connection. Also, the existence of invariant Ricci solitons with a non-conformal Killing vector field was proved for the such case. Moreover, a part of the proofs was obtained using the analytical calculation software packages. In this paper, we investigate invariant Ricci solitons on three-dimensional nonunimodular Lie groups with a left-invariant Riemannian metric and a semisymmet-ric connection. Analytical proofs of all theorems completing the classification of such solitons are presented.