For citation:
Mozhey N. P. On the Geometry of Three-dimensional Pseudo-Riemannian Homogeneous Spaces. II. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 2, pp. 172-184. DOI: 10.18500/1816-9791-2020-20-2-172-184, EDN: UGNADX
On the Geometry of Three-dimensional Pseudo-Riemannian Homogeneous Spaces. II
The problem of establishing links between the curvature and the topological structure of a manifold is one of the important problems of the geometry. In general, the purpose of the research of manifolds of various types is rather complicated. Therefore, it is natural to consider this problem in a narrower class of pseudo-Riemannian manifolds, for example, in the class of homogeneous pseudo-Riemannian manifolds. This paper is a continuation of the part I. 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, Einstein, Ricci-parallel, locally symmetric, conformally flat space are defined. In this paper, for all threedimensional pseudo-Riemannian homogeneous spaces, it is determined under what conditions the space is Ricci-flat, Einstein, Ricci-parallel, locally symmetric or conformally flat. In addition, for all these spaces, Levi – Cevita connections, curvature and torsion tensors, holonomy algebras, scalar curvatures, Ricci tensors are written out in explicit form. The results can find applications in mathematics and physics, since many fundamental problems in these fields are reduced to the study of invariant objects on homogeneous spaces.
- Besse A. Mnogoobraziya Eynshteyna [Einstein Manifolds: in 2 vols]. Moscow, Mir, 1990, vol. 1, 318 p.; vol. 2, 384 p. (in Russian).
- 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.
- 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).
- 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
- Alekseevsky D. V., Kimelfeld B. N. Classification of homogeneous conformally flat Riemannian manifolds. Math. Notes, 1978, vol. 24, no. 1, pp. 559–562.
- 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
- 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
- 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).
- 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
- Onishchik A. L. Topologiya tranzitivnykh grupp Li preobrazovaniy [Topology of transitive transformation groups]. Moscow, Fizmatlit, 1995. 384 p. (in Russian).
- Kobayashi S., Nomizu K. Foundations of differential geometry: in 2 vols. New York, John Wiley and Sons,1963, vol. 1, 330 p.; 1969, vol. 2, 448 p.
- 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
- 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
- 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).
- 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.
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