REDUCTIVE SPACES WITH A SOLVABLE GROUP OF TRANSFORMATIONS THAT DO NOT ADMIT EQUIAFFINE CONNECTIONS

UDC 514.765.1

Mozhey Natalya Pavlovna − PhD (Physics and Mathematics), Associate Professor, Assistant Professor, the Department of Software for Information Technologies. Belarusian State University of Informatics and Radioelectronics (6 P. Brovki str., 220013, Minsk, Republic of Belarus). E-mail: mozheynatalya@mail.ru

DOI: https://doi.org/ 10.52065/2520-6141-2025-290-3.

Key words: group of transformations, reductive space, normal connection, Ricci tensor, equiaffine connection.

For citation: Mozhey N. P. Reductive spaces with a solvable group of transformations that do not admit equiaffine connections. Proceedings of BSTU, issue 3, Physics and Mathematics. Informatics, 2025, no. 1 (290), pp. 16–19 (In Russian). DOI: 10.52065/2520-6141-2025-290-3.

Abstract

In the introduction, the object of research is indicated – connections on reductive spaces. If a Lie group acts transitively on a manifold, such manifold is called the homogeneous space, if a homogeneous space is reductive, then it always admits invariant connection. If there exists at least one invariant affine connection, then the space is isotropically-faithful. The aim of the work is to study three-dimensional reductive homogeneous spaces that do not admit equiaffine connections. The case of isotropically-faithful spaces on which a solvable group of transformations operates is considered. For three-dimensional reductive homogeneous spaces admitting normal connection, the question is studied under what conditions this space does not admit equiaffine connections. The basic notions are defined: homogeneous space, affine (invariant) connection, torsion tensor, curvature tensor, Ricci tensor, reductive space, holonomy algebra, normal connection, equiaffine connection. In the main part of the work, three-dimensional reductive homogeneous spaces that admit normal connection but do not admit equiaffine are found and explicitly presented, which is equivalent to describing the corresponding effective pairs of Lie algebras. Additionally, the invariant connections and Ricci tensors themselves are written out.

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References

  1. Kobayasi Sh., Nomidzu K. Osnovy differentsial’noy geometrii: v 2 t. [Foundations of differential geometry: in 2 vol.]. Moscow, Nauka Publ., 1981. 2 vol. (In Russian).
  2. Wang H. C. On invariant connections over a principal fibre bundle. Nagoya Math. J., 1958, no. 13, pp. 1–19.
  3. Nomizu K., Sasaki T. Affine differential geometry. Cambridge, Cambridge Univ. Press Publ., 1994. 263 p.
  4. Mozhey N. P. Three-dimensional reductive spaces of solvable Lie groups. Izvestiya Gomel'skogo gosudarstvennogo universiteta [Izvestiya Gomel State University], 2016, no. 6 (99), pp. 74–81 (In Russian).
  5. Kartan E. Rimanova geometriya v ortogonal’nom repere [Riemannian geometry in an orthogonal frame]. Moscow, Moskovskiy universitet Publ., 1960. 307 p.

15.10.2024