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 aim of this work is to describe perfect holonomy algebras of trivial connections on homogeneous spaces. We have concerned the case of the unsolvable 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 unsolvable 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 results obtained can be used in the study of manifolds, as well as have applications in various fields of geometry, topology, differential equations, analysis, algebra, in general relativity, in nuclear physics, elementary particle physics, etc., since many fundamental problems in these areas related to the study of homogeneous spaces and structures on them.