Open AccessOn Invariant Semisymmetric Connections on Three-Dimensional Non-Unimodular Lie Groups with the Metric of the Ricci Soliton
Author(s) -
D.V. Vylegzhanin,
Павел Николаевич Клепиков,
Евгений Дмитриевич Родионов,
Олеся Павловна Хромова
Publication year - 2021
Publication title -
izvestiâ altajskogo gosudarstvennogo universiteta
Language(s) - English
Resource type - Journals
eISSN - 1561-9451
pISSN - 1561-9443
DOI - 10.14258/izvasu(2021)4-13
Subject(s) - mathematics , ricci curvature , unimodular matrix , lie group , invariant (physics) , ricci flow , pure mathematics , einstein , mathematical physics , mathematical analysis , geometry , curvature
Metric connections with vector torsion, or semisymmetric connections, were first discovered by E. Cartan. They are a natural generalization of the Levi-Civita connection. The properties of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano, and other mathematicians.
Ricci solitons are the solution to the Ricci flow and a natural generalization of Einstein's metrics. In the general case, they were investigated by many mathematicians, which was reflected in the reviews by H.-D. Cao, R.M. Aroyo — R. Lafuente. This question is best studied in the case of trivial Ricci solitons, or Einstein metrics, as well as the homogeneous Riemannian case.
This paper investigates semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional non-unimodularLie groups with the left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that there are nontrivial invariant semisymmetric connections in this case. In addition, it is shown that there are nontrivial invariant Ricci solitons.