Abstract

In the introduction, an object of research is indicated – the holonomy algebras of affine connections on homogeneous spaces. The basic notions, such as an invariant affine connection, torsion and curvature tensors, a holonomy algebra are defined. The purpose of the work is the local classification of three-dimensional homogeneous spaces, admits the trivial affine connection perfect holonomy algebra only. We have concerned the case of the solvable Lie group of transformations. In the main part of the work a local description of three-dimensional homogeneous spaces, admitting only trivial affine connections with the perfect holonomy algebra, on which an solvable Lie group of transformations acts, is given. It is equivalent to describing the corresponding effective pairs of Lie algebras. The curvature tensors and the perfect holonomy algebras of the indicated connections are described explicitly. Studies are based on the use of properties of the Lie algebras, Lie groups and homogeneous spaces and they mainly have local character. The peculiarity of techniques presented in the work is the application of purely algebraic approach to the description of manifolds and structures on them, as well as compound of methods of differential geometry, the theory of Lie groups and algebras and the theory of homogeneous spaces.