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

### Abstract

A metric connection with vectorial torsion, or a semi-symmetric metric connection, was discovered by E. Cartan. Later, many mathematicians studied the properties of this connection. For example, K. Yano, I. Agricola and other mathematicians investigated the properties of the curvature tensor, geodesic lines, and also the behavior of the connection under conformal deformations of the original metric. In this paper, we study the Einstein equation on three-dimensional locally homogeneous (pseudo)Riemannian manifolds with metric connection with invariant vectorial torsion. A theorem is obtained stating that all such manifolds are either Einstein manifolds with respect to the Levi-Civita connection or conformally flat. Earlier, the Einstein equation in the case of three-dimensional locally symmetric (pseudo)Riemannian manifolds have been investigated by the authors.

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*Mathematical notes of NEFU*, 28(4), pp. 30-47. doi: https://doi.org/10.25587/SVFU.2021.26.84.003.

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