English Russian
Известия
Izvestiya of Altai State
University Journal
The News of Altai State University

 Архив журнала «Известия АлтГУ», начиная с 2017 г., размещен на новой версии сайта http://izvestiya.asu.ru
 Актуальная информация о журнале размещена на новой версии сайта http://izvestiya.asu.ru

Print ISSN 1561-9443
On-line ISSN 1561-9451
Issues list
Table of Contents
Physical science
Mathematics
About the Journal
Editorial board and Editorial advisory board
Regulations on reviewing research papers
Rules of the articles representation
Publication Ethics of the journal «Izvestiya of Altai State University»
 
1-2(85)2015
  MATHEMATICS

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.

DOI 10.14258/izvasu(2015)1.2-21

Key words: Lie group, Lie algebra, invariant Ricci soliton, left-invariant Riemannian metric, J. Milnor’s generalized bases

Full text at PDF, 612Kb. Language: Russian.

KLEPIKOV P.N.
Altai State University (Barnaul, Russia)
E-mail: askingnetbarnaul@gmail.com

OSKORBIN D.N.
Altai State University (Barnaul, Russia)
E-mail: oskorbin@yandex.ru

 

Print Edition of "Izvestiya of Altai State University" © 1996-2017 Altai State University.
All rights reserved. Any of parts of a journal or edition as a whole cannot be reprinted without the written sanction of the authors or publisher. On purchase of a journal to address to ASU publishing house:
Altai State University. 656049, 66 Dimitrova street, Barnaul, Russia. Telephone + 7 (3852) 366351.