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Mathematics. Mechanics. Informatics

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Mozhey N. P. Connections of Nonzero Curvature on Three-dimensional Non-reductive Spaces. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2017, vol. 17, iss. 4, pp. 381-393. DOI: 10.18500/1816-9791-2017-17-4-381-393, EDN: ZXJPLV

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Connections of Nonzero Curvature on Three-dimensional Non-reductive Spaces

Mozhey Natalya Pavlovna, Belarussian State University of Informatics and Radioelectronics

When a homogeneous space admits an invariant affine connection? If there exists at least one invariant connection then the space is isotropy-faithful, but the isotropy-faithfulness is not sufficient for the space in order to have invariantconnections. If a homogeneousspace is reductive, then the space admits an invariant connection.Thepurposeoftheworkisadescriptionofthree-dimensionalnon-reductivehomogeneousspaces, admitting invariant affine connections of nonzero curvature only, and the affine connections, curvature and torsion tensors. The basic notions, such as an isotropically-faithful pair, an affine connection, curvature and torsion tensors, a reductive space are defined. The local description of three-dimensional non-reductive homogeneous spaces, admitting connections of nonzero curvature only, is given. The local classification of such spaces is equivalent to the description of the effective pairs of Lie algebras. All invariant affine connections on those spaces are described, curvature and torsion tensors are found.

  1. Lie S., Engel F. Theorie der Transformationsgruppen. Leipzig, Teubner, 1893, vol. 3. 830 p.
  2. Gorbatsevich V. V., Onishchik A. L. Lie groups of transformations. Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. Мoscow, VINITI, 1988, vol. 20, pp. 103–240 (in Russian).
  3. Alekseevskii D. V., Vinogradov A. M., Lychagin V. V. Basic ideas and concepts of differ- ential geometry. Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. Мoscow, VINITI, 1988, vol. 28, pp. 5–297 (in Russian).
  4. Mozhey N. P. Trekhmernye izotropno-tochnye odnorodnye prostranstva i sviaznosti nanikh [Three-dimensional isotropy-faithful homogeneous spaces and connections on them]. Kazan’, Kazan’ Univ. Press, 2015. 394 p. (in Russian).
  5. Petrov A. Z. Novyye metody v obshchey teorii otnositel’nosti [New methods in the general theory of relativity]. Moscow, Nauka, 1966. 496 p. (in Russian).
  6. Onishchik A. L. Topologiya tranzitivnykh grupp Li preobrazovaniy [Topology of transitive Lie transformation groups]. Moscow, Fizmatlit, 1995. 384 p. (in Russian).
  7. Nomizu K. Invariant affine connections on homogeneous spaces. Amer. J. Math., 1954, vol. 76, no. 1, pp. 33–65.
  8. Kobayashi S., Nomizu K. Foundations of differential geometry. New York, John Wiley and Sons, 1963, vol. 1; 1969, vol. 2.
  9. Mozhey N. Three-dimensional Homogeneous Spaces, Not Admitting Invariant Connections. Izv. Saratov Univ. Ser. Math. Mech. Inform., 2016, vol. 16, iss. 4, pp. 413–421 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2016-16-4-413-421.
  10. Rashevski P. K. Simmetricheskiye prostranstva affinnoy svyaznosti s krucheniyem [Symmetric spaces of affine connection with torsion]. Trudy seminara po vektornomu i ten- zornomu analizu [Proceedings of the seminar on vector and tensor analysis], 1969, vol. 8, pp. 82–92 (in Russian).
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