Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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


For citation:

Mozhey N. P. On the Geometry of Three-dimensional Pseudo-Riemannian Homogeneous Spaces. I. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 1, pp. 29-41. DOI: 10.18500/1816-9791-2020-20-1-29-41, EDN: RRRHIO

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
02.03.2020
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Russian
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Article
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514.765
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RRRHIO

On the Geometry of Three-dimensional Pseudo-Riemannian Homogeneous Spaces. I

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

The problem of establishing links between the curvature and the topological structure of a manifold is one of the important problems of geometry. In general, the purpose of the research of manifolds of various types is rather complicated. Therefore, it is natural to consider this problem for a narrower class of pseudo-Riemannian manifolds, for example, for the class of homogeneous pseudo-Riemannian manifolds. The basic notions — such as an isotropically-faithful pair, a pseudo-Riemannian homogeneous space, an affine connection, curvature and torsion tensors, Levi – Cevita connection, Ricci tensor, Ricci-flat space, Einstein space, Ricci-parallel space, locally-symmetric space, conformally-flat space — are defined. In this paper, for all threedimensional Riemannian homogeneous spaces, it is determined under what conditions the space is Ricci-flat, Einstein, Ricci-parallel, locally-symmetric or conformally-flat. In addition, Levi – Cevita connections, curvature and torsion tensors, holonomy algebras, scalar curvatures, Ricci tensors are written out in explicit form for all these spaces. The results can be applied in mathematics and physics, since many fundamental problems in these fields are reduced to the study of invariant objects on homogeneous spaces.  

References: 
  1. Besse A. Mnogoobraziya Eynshteyna [Einstein Manifolds: in 2 vols]. Moscow, Mir, 1990. Vol. 1, 318 p.; vol. 2, 384 p. (in Russian).
  2. Wang M. Einstein metrics from symmetry and Bundle Constructions. In: Surveys in Differential Geometry. VI: Essays on Einstein Manifolds. Boston, MA, International Press, 1999, pp. 287–325.
  3. Reshetnyak Yu. G. Isothermal coordinates in manifolds of bounded curvature. Sib. Matem. Zhurn., 1960, vol. 1, no. 1, pp. 88–116; vol. 1, no. 2, pp. 248–276 (in Russian).
  4. Gray A. Einstein-like manifolds which are not Einstein. Geom. Dedicata, 1978, vol. 7, iss. 3, pp. 259–280. DOI: https://doi.org/10.1007/BF00151525
  5. Alekseevsky D. V., Kimelfeld B. N. Classification of homogeneous conformally flat Riemannian manifolds. Math. Notes, 1978, vol. 24, no. 1, pp. 559–562
  6. Kowalski O., Nikcevic S. On Ricci eigenvalues of locally homogeneous Riemann 3-manifolds. Geom. Dedicata, 1996, vol. 62, pp. 65–72. DOI: https://doi.org/10.1007/BF00240002
  7. Rodionov E. D., Slavsky V. V., Chibrikova L. N. Locally conformally homogeneous pseudo-Riemannian spaces. Sib. Adv. Math., 2007, vol. 17, pp. 186–212. DOI: https://doi.org/10.3103/S1055134407030030
  8. Rodionov E. D. Compact simply connected standard homogeneous Einstein manifolds with holonomy group SO(n). Izvestiya Altayskogo gosudarstvennogo universiteta [Izvestiya of Altai State University], 1997, no. 1 (3), pp. 7–10 (in Russian).
  9. Nikonorov Yu. G., Rodionov E. D., Slavsky V. V. Geometry of homogeneous Riemannian manifolds. J. Math. Sci., 2007, vol. 146, pp. 6313–6390. DOI: https://doi.org.10.1007/s10958-007-0472-z
  10. Onishchik A. L. Topologiya tranzitivnykh grupp Li preobrazovaniy [Topology of transitive transformation groups]. Moscow, Fizmatlit, 1995. 384 p. (in Russian).
  11. Kobayashi S., Nomizu K. Foundations of differential geometry: in 2 vols. New York, John Wiley and Sons, vol. 1, 1963. 330 p.; vol. 2, 1969. 448 p.
  12. Mozhey N. P. Affine connections on three-dimensional pseudo-Riemannian homogeneous spaces. I. Russ. Math., 2013, vol. 57, iss. 12, pp. 44–62. DOI: https://doi.org/10.3103/S1066369X13120050
  13. Mozhei N. P. Affine connections on three-dimensional pseudo-Riemannian homogeneous spaces. II. Russ. Math., 2014, vol. 58, iss. 6, pp. 28–43. DOI: https://doi.org/10.3103/S1066369X14060048
  14. Mozhey N. P. Trekhmernye izotropno-tochnye odnorodnye prostranstva i sviaznosti na nikh [Three-dimensional isotropy-faithful homogeneous spaces and connections on them]. Kazan, KFU Publishing House, 2015. 394 p. (in Russian).
  15. Garcia A., Hehl F. W., Heinicke C., Macias A. The Cotton tensor in Riemannian spacetimes. Classical and Quantum Gravity, 2004, vol. 21, no. 4, pp. 1099–1118.
Received: 
03.11.2018
Accepted: 
31.01.2019
Published: 
02.03.2020