Three-dimensional homogeneous spaces of nonsolvable lie groups with equiaffine connections of nonzero curvature

Abstract

The problem of establishing links between the curvature and the structure of a manifold is one of the important problems in geometry. The purpose of the work is a local description of the equiaffine (locally equiaffine) connections on three-dimensional homogeneous spaces that only admit invariant connections of nonzero curvature. We have considered the case of the nonsolvable Lie group of transformations. Basic notions are defined, such as an isotropically-faithful pair, an (invariant) affine connection, curvature and torsion tensors, Ricci tensor, equiaffine (locally equiaffine) connection. In the main part of the work, for threedimensional homogeneous spaces of nonsolvable Lie groups (that only admit invariant connections of nonzero curvature) equiaffine (locally equiaffine) connections and their Ricci tensors are found and written out in explicit form. A special feature of the methods presented in the work is an application of a purely algebraic approach to a description of manifolds and structures on them. Results obtained in the work can be applied to solving problems in differential geometry, differential equations, topology, as well as in other areas of mathematics and physics. The algorithms for finding connections can be computerized and used to solve similar problems in large dimensions.