Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Mozhey N. P. Homogeneous spaces of unsolvable Lie groups that do not admit equiaffine connections of nonzero curvature. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2023, vol. 23, iss. 4, pp. 435-442. DOI: 10.18500/1816-9791-2023-23-4-435-442, EDN: DRMGUT

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
30.11.2023
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Russian
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Article
UDC: 
514.76
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DRMGUT

Homogeneous spaces of unsolvable Lie groups that do not admit equiaffine connections of nonzero curvature

Autors: 
Mozhey Natalya Pavlovna, Belarussian State University of Informatics and Radioelectronics
Abstract: 

An important subclass among homogeneous spaces is formed by isotropically-faithful homogeneous spaces, in particular, this subclass contains all homogeneous spaces admitting invariant affine connection. An affine connection is equiaffine if it admits a parallel volume form. The purpose of the work is the local description of the three-dimensional homogeneous spaces that do not admit invariant equiaffine connections of nonzero curvature. We have concerned the case of the unsolvable Lie group of transformations. The basic notions, such as an isotropically-faithful pair, an invariant affine connection, curvature and torsion tensors, Ricci tensor, equiaffine connection are defined. A local study of homogeneous spaces is equivalent to the investigation of pairs consisting of a Lie algebra and its subalgebra. For three-dimensional homogeneous  spaces of nonsolvable Lie groups that admit invariant connections of nonzero curvature only, it is determined under what conditions the space does not admit  equiaffine connections. Studies are based on the use of properties of the Lie algebras, Lie groups and homogeneous spaces and they mainly have local character. A feature of the methods presented in the work is the application of a purely algebraic approach to the description of homogeneous spaces and connections on them. The results obtained in the work can be used in works on differential geometry, differential equations, topology, as well as in other areas of mathematics and physics, since many fundamental problems in these areas relate to the investigation of invariant objects on homogeneous spaces, the algorithms can be computerized and used for the solution of similar problems in large dimensions.

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Received: 
23.08.2022
Accepted: 
22.03.2023
Published: 
30.11.2023