EQUIAFFINE CONNECTIONS OF ZERO CURVATURE ON HOMOGENEOUS SPACES OF SOLUBLE GROUPS OF TRANSFORMATIONS

UDC 514.76

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/ 10.52065/2520-6141-2024-284-2.

Key words: equiaffine connection, transformation group, homogeneous space, curvature tensor, torsion tensor.

For citation: Mozhey N. P. Equiaffine connections of zero curvature on homogeneous spaces of soluble groups of transformations. Proceedings of BSTU, issue 3, Physics and Mathematics. Informatics, 2024, no. 2 (284), pp. 10–18 (In Russian). DOI: 10.52065/2520-6141-2024-284-2.

Abstract

In the introduction, the object of research is indicated – affine connections on homogeneous spaces. When a homogeneous space admits an invariant connection? If there exists at least one invariant affine connection, then the space is isotropically-faithful. In this article we study three-dimensional isotropicalyfaithful homogeneous spaces on which a solvable Lie group of transformations operates, allowing invariant connections of zero curvature only. The purpose of the work is to determine under what conditions these spaces do not admit equiaffine connections. The basic notions, such as isotropicallyfaithful pair, affine connection, curvature and torsion tensors, Ricci tensor, equiaffine connection are defined. In the main part of the paper, a complete description of three-dimensional homogeneous spaces with a solvable group of transformations, allowing invariant affine connections of zero curvature only, but not allowing equiaffine connections, is found and given explicitly. 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. The results obtained can be used in the study of manifolds, as well as have applications in various fields of mathematics and physics, since many fundamental problems in these fields are connected with the study of invariant objects on homogeneous spaces.

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References

  1. Blaschke W. Vorlesungen ȕber Differentialgeometrie. Berlin, Springer Publ., 1923, vol. 2. 230 s. (Auf Deutsch).
  2. Olver P. Recursive moving frames. Results Math., 2011, vol. 60, pp. 423–452.
  3. Veblen O., Whitehead J. The foundations of differential geometry. Cambridge, Cambridge Univ. Press Publ., 1932. 230 p.
  4. Mozhey N. P. Connections of zero curvature on homogeneous spaces of solvable Lie groups. Izvestiya Gomel'skogo gosudarstvennogo universiteta [Izvestiya Gomel State University], 2017, no. 6 (105), pp. 104– 111 (In Russian).
  5. Nomizu K., Sasaki T. Affine differential geometry. Cambridge, Cambridge Univ. Press Publ., 1994. 263 p.

 15.03.2024