Three-Dimensional Metric Lie Groups with Vectorial Torsion and Zero Curvature Tensor
УДК 517.795
Abstract
Recently, the study of (pseudo)Riemannian manifolds with different metric connections different from the Levi-Civita connection becomes relevant.A metric connection with vectorial torsion (also known as a semi-symmetric connection) is one of the often considered connections.The connection 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.Moreover, this connection plays an important role in the case of two-dimensional surfaces since, in this case, any metric connection is a connection with vectorial torsion.Thus, the problem of studying (pseudo)Riemannian manifolds with metric connection with vectorial torsion, the curvature tensor of which is zero, is arisen.This paper is devoted to solving the problem in the case of three-dimensional metric Lie groups. In addition, a mathematical model is presented that allows one to calculate the components of the curvature tensor of a metric connection with vectorial torsion in the case of metric Lie groups.
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References
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