On the maximal dimension of Lie algebras of infinitesimal affine transformations of direct products of more than two spaces of affine connection of the first type
Abstract
In modern differential geometry, one of the main problems of the geometry of a space with a differential-geometric structure is the study of the group of affine transformations (automorphisms) of this space. The studies of automorphisms in various spaces of affine connections are devoted to the works of E. Cartan, P. K. Rashevsky, P. A. Shirokov, I. P. Egorov, A.Ya. Sultanov and other scientists.
Affine conversions in direct products of two spaces with affine connection were considered in the works of M. V. Morgun. In the case of direct products of more than two spaces with affine connection, the question of affine envelopes, these spaces are stable.
In the article Glebova M. V. and Sultanov A.Ya an estimate was obtained for the dimension of the Lie algebra of infinitesimal affine transformations of spaces with affine connection that represent a direct product of at least three non-projective Euclidean spaces of the special condition. Such spaces are called spaces of the first type.
In this paper, the accuracy of this estimate is proven. To solve the problem, a system of linear homogeneous equations is investigated, which is satisfied by the components of an arbitrary infinitesimal affine transformation.
This system is obtained using the properties of the Lie derivative applied to the tensor field of curvature of the spaces under consideration. An estimate of the rank of this system made it possible to obtain a lower estimate for the rank of the matrix of the original system.
Reference
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