Non-reductive spaces with equiaffine connections of nonzero curvature

N. P. Mozhey

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 3, pp. 305-316.

Résumé

The introduction of this article states the object of our investigation which is structures on homogeneous spaces. The problem of establishing links between the curvature and the structure of a manifold is one of the important problems of geometry. In general, the research of manifolds of various types is rather complicated. Therefore, it is natural to consider this problem in a narrower class of non-reductive homogeneous spaces. If a homogeneous space is reductive, then the space admits an invariant connection. If there exists at least one invariant connection, then the space is isotropy-faithful. This work studies three-dimensional non-reductive homogeneous spaces that admit invariant affine connections of nonzero curvature only. The basic notions, such as an isotropically-faithful pair, an (invariant) affine connection, curvature and torsion tensors, Ricci tensor, an equiaffine (locally equiaffine) connection, and a reductive space are defined. The purpose of this work is the description of equiaffine (locally equiaffine) connections on such spaces. In the main part of this paper, for three-dimensional non-reductive homogeneous spaces (that admit invariant connections of nonzero curvature only) equiaffine (locally equiaffine) connections are found and written out in explicit form. The features of the methods presented in the work is the application of a purely algebraic approach to the description of manifolds and structures on them. In the conclusion, the results obtained in the work are indicated. The results can be used in works on differential geometry, differential equations, topology, as well as in other areas of mathematics and physics. The algorithms for finding connections can be computerized and used for the solution of similar problems in large dimensions.

Bibliographie

[1] Skott P., Geometries on Three-dimensional Manifolds, Mir, M., 1986, 163 pp. (in Russian)

[2] Cartan E., La géométrie des espaces de Riemann, Mémorial des sciences mathématiques, 9, Gauthier-Villars et C$^{\textrm{o}}$, Paris, 1925, 64 pp. (in French)

[3] Alekseyevskiy D. V., Vinogradov A. M., Lychagin V. V., “Basic ideas and concepts of differential geometry”, Itogi Nauki i Tekhniki. Seriya: Sovremennye Problemy Matematiki. Fundamental'nye Napravleniya, 28, VINITI, M., 1988, 5–289 (in Russian)

[4] Mozhey N. P., “Connections of nonzero curvature on three-dimensional non-reductive spaces”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 17:4 (2017), 381–393 (in Russian) | DOI | Zbl

[5] Onishchik A. L., Topology of Transitive Transformation Groups, Fizmatlit, M., 1995, 384 pp. (in Russian)

[6] Nomizu K., “Invariant affine connections on homogeneous spaces”, American Journal of Mathematics, 76:1 (1954), 33–65 | DOI | Zbl

[7] Kobayashi S., Nomizu K., Foundations of Differential Geometry, in 2 vols, v. 2, John Wiley and Sons, New York, 1969, 488 pp. | Zbl

[8] Nomizu K., Sasaki T., Affine Differential Geometry: Geometry of Affine Immersions, Cambridge Univ. Press, Cambridge–New York, 1994, 264 pp.

[9] Mozhey N. P., “Three-dimensional homogeneous spaces, not admitting invariant connections”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 16:4 (2016), 413–421 (in Russian) | DOI | Zbl

[10] Rashevski P. K., “Symmetric spaces of affine connection with torsion”, Proceedings of the Seminar on Vector and Tensor Analysis, 8, 1969, 82–92 (in Russian)