Three-dimensional non-reductive homogeneous spaces of solvable groups Lie, admitting affine connections

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 non-reductive homogeneous spaces, admitting invariant affine connections. We concerned only case, when Lie group is solvable. The local classification of homogeneous spaces is equivalent to the description of effective pairs of Lie algebras. The peculiarity of techniques presented in the work is the application of purely algebraic approach, the compound of different methods of differential geometry, theory of Lie groups, Lie algebras and homogeneous spaces.