THREE-DIMENSIONAL NON-REDUCTIVE HOMOGENEOUS SPACES OF UNSOLVABLE LIE GROUPS

Abstract

The purpose of the work is a description of three-dimensional non-reductive homogeneous spaces that allow invariant affine connections together with their curvature and torsion tensors, holonomy algebras. We are concerned with the case, when Lie group of transformations is unsolvable and a stabilizer is unsolvable too. An object of investigation is concerned with non-reductive spaces and connections on them. The basic notions, such as an isotropically-faithful pair, a reductive space, an affine connection, curvature and torsion tensors, and holonomy algebra are defined. The local description of three-dimensional non-reductive homogeneous spaces with unsolvable Lie group of transformations and an unsolvable stabilizer, which allow affine connections, is given. The local classification of homogeneous spaces is equivalent to the description of the effective pairs of 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 Lie algebras and groups, as well as homogeneous spaces and they are mainly local in character.  The features of the methods presented in the work is the application of a purely algebraic approach to the description of homogeneous spaces and connections on them, as well as the combination of methods of differential geometry, the theory of Lie groups and algebras and the theory of homogeneous spaces. The results can be used in the study of manifolds and can find application in various areas of mathematics and physics, since many fundamental problems in these areas relate to the investigation of invariant objects on homogeneous spaces.