Three-Dimensional Locally Symmetric (Pseudo)Riemannian Manifolds with Vectorial Torsion and Zero Curvature Tensor
УДК 514.764.2
Abstract
A metric connection with vectorial torsion (also known as a semi-symmetric connection) is one of the three main connections described by E. Cartan. 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.K. Yano proved an important theorem on the connection of conformal deformations and metric connections with vectorial torsion. Namely, a Riemannian manifold admits a metric connection with vectorial torsion, the curvative tensor of which is zero, if and only if it is conformally flat. 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 locally symmetric manifolds. 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 locally homogeneous (pseudo)Riemannian manifolds.
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References
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