CONNECTIONS ON NON-REDUCTIVE HOMOGENEOUS SPACES WITH AN UNSOLVABLE GROUP OF TRANSFORMATIONS

Abstract

When does a homogeneous space allow an invariant affine connection? If at least one invariant connection exists, then the space is isotropy-faithful, but the isotropy-faithfulness is insufficient for the space in order to have invariant connections. If a homogeneous space is reductive, then the space allows an invariant connection. The purpose of the work is to describe invariant affine connections on three-dimensional non-reductive homogeneous spaces together with their curvature and torsion tensors, holonomy algebras. We concerned the case, when the Lie group of transformations is unsolvable and a stabilizer is solvable. The basic notions, such as an isotropy-faithful pair, a reductive space, an affine connection, curvature and torsion tensors, holonomy algebra are defined. A local description of three-dimensional non-reductive homogeneous spaces with an unsolvable Lie group of transformations and a solvable stabilizer, allowing affine connections, is given. A local classification of homogeneous spaces is equivalent to that of the effective pairs of the Lie algebras. All invariant affine connections on those spaces are described, curvature and torsion tensors, holonomy algebras are found. Studies are based on the use of properties of the Lie algebras, Lie groups and homogeneous spaces. They are mainly local in character.