On the Symmetric Einstein Equation for Three-Dimensional Lie Groups with Left-Invariant Riemannian Metric and Semi-Symmetric Connection

УДК 514.765

  • A.A. Pavlova Altai State University (Barnaul, Russia) Email: anya.0596@mail.ru
  • O.P. Khromova Altai State University (Barnaul, Russia) Email: khromova.olesya@gmail.com
  • E.D. Rodionov Altai State University (Barnaul, Russia) Email: edr2002@mail.ru
  • D.V. Vylegzhanin Belarusian State University (Minsk, Belarus) Email: Vylegzhanin@bsu.by
Keywords: semisymmetric connections, Einstein metrics, Lie groups, left-invariant Riemannian metrics

Abstract

Riemannian manifolds with a Levi-Civita connection and constant Ricci curvature, or Einstein manifolds, were studied in the works of many mathematicians. This question has been most studied in the homogeneous Riemannian case. In this direction, the most famous ones are the results of works by D.V. Alekseevsky, M. Wang, V. Ziller, G. Jensen, H.Laure, Y.G. Nikonorov, E.D. Rodionov and other mathematicians. At the same time, the question of studying Einstein manifolds has been little studied for the case of an arbitrary metric connection. This is primarily due to the fact that the Ricci tensor of metric connection is not, generally speaking, symmetric.

In this paper, we consider semisymmetric connections on 3-dimensional Lie groups with a left-invariant Riemannian metric. For the symmetric part of the Ricci tensor, the Einstein equation is studied. As a result of the research carried out, a classification of 3-dimensional metric Lie groups and the corresponding semisymmetric connections in the case of the Einstein symmetric equation has been obtained.

Earlier in the works of P.N. Klepikov, E.D. Rodionov and O.P. Khromova, the classical Einstein equation was studied, and it was proved that if the classical.

The Einstein equation holds for a 3-dimensional Lie group with left-invariant (pseudo) Riemannian metric and semi-symmetric connection. Then, either the connection is a Levi-Civita connection, or the curvature tensor of the connection is equal to zero.

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Author Biographies

A.A. Pavlova, Altai State University (Barnaul, Russia)

аспирантка Института математики и информационных технологий

O.P. Khromova, Altai State University (Barnaul, Russia)

кандидат физико-математических наук, доцент, доцент кафедры математического анализа

E.D. Rodionov, Altai State University (Barnaul, Russia)

доктор физико-математических наук, профессор, профессор кафедры математического анализа

D.V. Vylegzhanin, Belarusian State University (Minsk, Belarus)

кандидат физико-математических наук, доцент кафедры геометрии, топологии и методики преподавания математики

References

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Клепиков П.Н., Родионов Е.Д., Хромова О.П. Уравнение Эйнштейна на трехмерных локально симметрических (псевдо)римановых многообразиях с векторным кручением // Математические заметки СВФУ. 2019. Т. 26. № 4. DOI: 10.25587/SVFU.2019.49.61.003.

Published
2022-09-09
How to Cite
Pavlova A., Khromova O., Rodionov E., Vylegzhanin D. On the Symmetric Einstein Equation for Three-Dimensional Lie Groups with Left-Invariant Riemannian Metric and Semi-Symmetric Connection // Izvestiya of Altai State University, 2022, № 4(126). P. 140-143 DOI: 10.14258/izvasu(2022)4-21. URL: http://izvestiya.asu.ru/article/view/%282022%294-21.
Section
Математика и механика