Non-reductive homogeneous spaces not admitting normal connections

N. P. Mozhey

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 3, pp. 284-296

Résumé

The purpose of the work is the classification of three-dimensional non-reductive homogeneous spacesnot admitting normal connections, affine connections, their torsion tensors, curvature and holonomy algebras. The object of investigation arepointed-non-reductive spaces and connections on them. The basic notions, such as the isotropically-faithful pair, reductive space, af?ne connection, curvature tensor and torsion tensor, holonomy algebra and normal connection are defined. The local study of homogeneous spaces is equivalent to the investigation of pairs consisting of the Lie algebra and its subalgebra. The local classification of three-dimensional non-reductive homogeneous spaces with the unsolvable Lie group of transformations not admitting normal connections is given. All invariant affine connections on those spaces are described, curvature and torsion tensors are found; the holonomy algebras are investigated and it has been determined that the invariant connection is not normal. Studies are based on the use of properties of the Lie algebras, Lie groups and homogeneous spaces and they mainly have local character. The characteristic of the method 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 obtained 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.