Reductive homogeneous spaces and connections on them

Abstract

When a homogeneous space admits an invariant affine connection? If there exists at least one invariant connection then the space is isotropy-faithful, but the isotropy-faithfulness is not sufficient for the space in order to have invariant connections. If a homogeneous space is reductive, then the space admits an invariant connection. The purpose of the work is the classification of three-dimensional reductive homogeneous spaces and invariant affine connections on them. We describe all local three-dimensional reductive homogeneous spaces, it is equivalent to the description of effective pairs of Lie algebras, and all invariant affine connections on the spaces together with their curvature, torsion tensors and holonomy algebras. The results of work can be used in research work of the differential geometry, differential equations, topology, in the theory of representations, in the theoretical physics.