Einstein equation on three-dimensional locally symmetric (pseudo)Riemannian manifolds with vectorial torsion

Keywords: locally symmetric space, Lie algebra, vectorial torsion, invariant (pseudo)- Riemannian metric, Einstein manifold

Abstract

The study of (pseudo)Riemannian manifolds with different metric connections different from the Levi-Civita connection has become a subject of current interest lately. A metric connection with vectorial torsion (also known as a semi-symmetric connection) is a frequently considered one of them. The correlation between the conformal deformations of Riemannian manifolds and metric connections with vectorial torsion on them was established in the works of K. Yano. Namely, a Riemannian manifold admits a metric connection with vectorial torsion, the curvature tensor of which is zero, if and only if it is conformally flat. In this paper, we study the Einstein equation on three-dimensional locally symmetric (pseudo)Riemannian manifolds with metric connection with invariant vectorial torsion. We obtain a theorem stating that all such manifolds are either Einstein manifolds with respect to the Levi-Civita connection or conformally flat.

References


[1]
Cartan E., “Sur les variétés à connexion affine et la théorie de la relativité généralisée. II,” Ann. Sci. Ec. Norm. Supér. (3), ´ 42, 17–88 (1925).

[2]
Muniraja G., “Manifolds admitting a semi-symmetric metric connection and a generalization of Schur’s theorem,” Int. J. Contemp. Math. Sci., 3, No. 25, 1223–1232 (2008).

[3]
Agricola I. and Thier C., “The geodesics of metric connections with vectorial torsion,” Ann. Global Anal. Geom., 26, 321–332 (2004).

[4]
Murathan C. and Özgür C., “Riemannian manifolds with a semi-symmetric metric connection satisfying some semisymmetry conditions,” Proc. Est. Acad. Sci., 57, No. 4, 210–216 (2008).

[5]
Yilmaz H. B., Zengin F. Ö., and Uysal. S. A., “On a semi-symmetric metric connection with a special condition on a Riemannian manifold,” Eur. J. Pure Appl. Math., 4, No. 2, 152–161 (2011).

[6]
Zengin F. Ö., Demirbaǧ S. A., Uysal. S. A., and Yilmaz H. B., “Some vector fields on a Riemannian manifold with semi-symmetric metric connection,” Bull. Iran. Math. Soc., 38, No. 2, 479–490 (2012).

[7]
Agricola I. and Kraus M., “Manifolds with vectorial torsion,” Differ. Geom. Appl., 46, 130–147 (2016).

[8]
Yano K., “On semi-symmetric metric connection,” Rev. Roum. Math. Pures Appl., 15, 1579–1586 (1970).

[9]
Barua B. and Ray A. Kr., “Some properties of a semi-symmetric metric connection in a Riemannian manifold,” Indian J. Pure Appl. Math., 16, No. 7, 736–740 (1985)

[10]
De U. C. and De B. K., “Some properties of a semi-symmetric metric connection on a Riemannian manifold,” Istanb. Üniv. Fen. Fak. Mat. Derg., 54, 111–117 (1995).

[11]
Besse A., Einstein Manifolds [in Russian], Mir, Moscow (1990).

[12]
Chaturvedi B. B. and Gupta B. K., “Study on semi-symmetric metric spaces,” Novi Sad J. Math., 44, No. 2, 183–194 (2014).

[13]
Sekigawa K., “On some 3-dimensional curvature homogeneous spaces,” Tensor, New Ser., 31, 87–97 (1977).

[14]
Calvaruso G., “Homogeneous structures on three-dimensional Lorentzian manifolds,” J. Geom. Phys., 57, 1279–1291 (2007).

[15]
Khromova O. P., “Application of symbolic computation packages to investigation of onedimensional curvature operator on non-reductive homogeneous pseudo-Riemannian manifolds [in Russian],” Izv. Altaysk. Gos. Univ., 1, 140–143 (2017).

[16]
Mozhey N. P., “Cohomology of three-dimensional homogeneous manifolds [in Russian],” Tr. Barnaulsk. Gos. Tekhn. Univ., 6, 13–18 (2014).
How to Cite
Klepikov, P., Rodionov, E. and Khromova, O. (2020) “Einstein equation on three-dimensional locally symmetric (pseudo)Riemannian manifolds with vectorial torsion”, Mathematical notes of NEFU, 26(4), pp. 25-36. doi: https://doi.org/10.25587/SVFU.2019.49.61.003.
Section
Mathematics