Geodesic
Einstein's equation on three-dimensional metric Lie groups with vector torsion
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Classical and Modern Geometry" Dedicated to the 100th Anniversary of the Birth of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 3, Tome 181 (2020), pp. 41-53.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we study the Einstein equation on three-dimensional Lie groups equipped with a left-invariant (pseudo) Riemannian metric and a metric connection with left-invariant vector torsion. We prove that all such Lie groups are either Einstein manifolds with respect to the Levi-Civita connection or conformally flat manifolds.
Keywords: Lie algebra, left-invariant (pseudo) Riemannian metric, Einstein manifold.
Mots-clés : Lie group, vector torsion 
@article{INTO_2020_181_a5,
     author = {P. N. Klepikov and E. D. Rodionov and O. P. Khromova},
     title = {Einstein's equation on three-dimensional metric {Lie} groups with vector torsion},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {41--53},
     publisher = {mathdoc},
     volume = {181},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_181_a5/}
}
TY  - JOUR
AU  - P. N. Klepikov
AU  - E. D. Rodionov
AU  - O. P. Khromova
TI  - Einstein's equation on three-dimensional metric Lie groups with vector torsion
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2020
SP  - 41
EP  - 53
VL  - 181
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2020_181_a5/
LA  - ru
ID  - INTO_2020_181_a5
ER  - 
%0 Journal Article
%A P. N. Klepikov
%A E. D. Rodionov
%A O. P. Khromova
%T Einstein's equation on three-dimensional metric Lie groups with vector torsion
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2020
%P 41-53
%V 181
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2020_181_a5/
%G ru
%F INTO_2020_181_a5
P. N. Klepikov; E. D. Rodionov; O. P. Khromova. Einstein's equation on three-dimensional metric Lie groups with vector torsion. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Classical and Modern Geometry" Dedicated to the 100th Anniversary of the Birth of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 3, Tome 181 (2020), pp. 41-53. http://geodesic.mathdoc.fr/item/INTO_2020_181_a5/

[1] Besse A., Mnogoobraziya Einshteina, Mir, M., 1990

[2] Rodionov E. D., Slavskii V. V., Chibrikova L. N., “Lokalno konformno odnorodnye psevdorimanovy prostranstva”, Mat. tr., 9:1 (2006), 130–168 | MR | Zbl

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

[4] Agricola I., Thier C., “The geodesics of metric connections with vectorial torsion”, Ann. Glob. Anal. Geom., 26 (2004), 321–332 | DOI | MR | Zbl

[5] Barua B., Ray A. Kr., “Some properties of a semi-symmetric metric connection in a Riemannian manifold”, Indian J. Pure Appl. Math., 16:7 (1985), 736–740 | MR | Zbl

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

[7] Cartan E., “Sur les variétés à connexion affine et la théorie de la relativité généralisée (deuxième partie)”, Ann. Ecole Norm. Super., 42 (1925), 17–88 | DOI | MR | Zbl

[8] Chaturvedi B. B., Gupta B. K., “Study on semi-symmetric metric spaces”, Novi Sad J. Math., 44:2 (2014), 183–194 | MR | Zbl

[9] Cordero L. A., Parker P. E., “Left-invariant Lorentzian metrics on 3-dimensional Lie groups”, Rend. Mat., 17 (1997), 129–155 | MR | Zbl

[10] De U. C., De B. K., “Some properties of a semi-symmetric metric connection on a Riemannian manifold”, Istanbul Univ. Fen. Fak. Mat. Der., 54 (1995), 111–117 | MR

[11] Milnor J., “Curvature of left invariant metric on Lie groups”, Adv. Math., 21 (1976), 293–329 | DOI | MR | Zbl

[12] Muniraja G., “Manifolds admitting a semi-symmetric metric connection and a generalization of Schur's theorem”, Int. J. Contemp. Math. Sci., 3:25 (2008), 1223–1232 | MR | Zbl

[13] Murathan C., Özgür C., “Riemannian manifolds with a semi-symmetric metric connection satisfying some semisymmetry conditions”, Proc. Eston. Acad. Sci., 57:4 (2008), 210–216 | DOI | MR | Zbl

[14] Yano K., “On semi-symmetric metric connection”, Rev. Roum. Math. Pure Appl., 15 (1970), 1579–1586 | MR | Zbl

[15] Yilmaz H. B., Zengin F. Ö., Uysal. S. A., “On a semi-symmetric metric connection with a special condition on a Riemannian manifold”, Eur. J. Pure Appl. Math., 4:2 (2011), 152–161 | MR

[16] Zengin F. Ö., Demirbağ S. A., Uysal. S. A., Yilmaz H. B., “Some vector fields on a Riemannian manifold with semi-symmetric metric connection”, Bull. Iran. Math. Soc., 38:2 (2012), 479–490 | MR | Zbl