Affine transformations of the tangent bundle with a complete lift connection over a manifold with a linear connection of special type

Abstract

The theory of tangent bundles over a differentiable manifold M be­longs to the geometry and topology of manifolds and is an intensively developing area of the theory of fiber spaces. The foundations of the theo­ry of fibered spaces were laid in the works of S. Eresman, A. Weil, A. Mo­ri­moto, S. Sasaki, K. Yano, S. Ishihara. Among Russian scientists, tangent bund­les were investigated by A. P. Shirokov, V. V. Vishnevsky, V. V. Shu­rygin, B. N. Shapukov and their students. In the study of automorphisms of generalized spaces, the question of infinitesimal transformations of connections in these spaces is of great importance. K. Yano, G. Vrancianu, P. A. Shirokov, I. P. Egorov, A. Z. Pet­rov, A. V. Aminova and others have studied movements in different spac­es. The works of K. Sato and S. Tanno are devoted to the motions and au­tomorphisms of tangent bundles. Infinitesimal affine collineations in tan­gent bundles with a synectic connection were considered by H. Shadyev. At present, the question of the motions of fibered spaces is considered in the works of A. Ya. Sultanov, in which infinitesimal transformations of a bundle of linear frames with a complete lift connection, the Lie algebra of holomorphic affine vector fields in arbitrary Weyl bundles are investi­gated. In this paper we obtain exact upper bounds for the dimensions of Lie algebras of infinitesimal affine transformations in tangent bundles with a synectic connection A. P. Shyrokov.

Keywords

• differentiable manifold
• tangent bundle
• synectic connection
• infinitesimal affine transformation
• lie algebra