Normal connections on three-dimensional homogeneous spaces with a non-solvable transformation group. I. A non-solvable stabilizer

In this paper we present a complete local classification of three-dimensional homogeneous spaces admitting a normal connection. We consider only the case of a non-solvable Lie group of transformations with a non-solvable stabilizer. The local classification of homogeneous spaces is equivalent to the description of effective pairs of Lie algebras. We describe all invariant affine connections on those homogeneous spaces together with their curvature and torsion tensors. We study the holonomy algebras of homogeneous spaces and find when the invariant connection is normal. We use an algebraic approach for describing the connections as well as methods of the theories of Lie groups, Lie algebras and homogeneous spaces.