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Klepikov, Pavel Nikolaevich; Rodionov, Evgeniĭ Dmitrievich; Khromova, Olesya Pavlovich

Einstein equation on three-dimensional locally symmetric (pseudo)Riemannian manifolds with vectorial torsion. (Russian. English summary) Zbl 07819531

Mat. Zamet. SVFU 26, No. 4, 25-36 (2019).
Summary: 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.

Cited in 1 Document

MSC:

53-XX Differential geometry
58-XX Global analysis, analysis on manifolds

Keywords:

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

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[17] Поступила в редакцию 1 августа 2019 г. После доработки 29 сентября 2019 г. Принята к публикации 27 ноября 2019 г.
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