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Mathematics

On Lie algebras of infinitesimal affine transformations in tangent bundles with a complete lift connection

G. A. Sultanova

Penza State University, Penza

Abstract: Background. The study of infinitesimal automorphisms of connections in the fiber spaces is one of the important problems in the theory of these spaces. The infinitesimal isometrics of tangent bundles were studied by S. Sasaki. Yano and Kobayashi considered the questions about the canonical decomposition of infinitesimal affine transformations. Among Russian scientists H. Shadyev saw movement in tangent bundles of the first order with synectic connection. In this paper we consider the tangent bundles with a complete lift connection, where the base of the bundle is the most moving two-dimensional space of affine connection. We study one of the types of two-dimensional spaces with affine connection obtained by I. P. Egorov whose movement groups have a maximum dimension of 4. We built algebra of infinitesimal automorphisms of spaces ($TM_2,\bigtriangledown ^{(0)}$) and clarified the question of solvability of this algebra. Materials and methods. The object of study is the space ($TM_2,\bigtriangledown ^{(0)}$). We are using the methods of the tensor calculus, the theory of Lie derivative. The variety, function, tensor fields are assumed to be smooth of class $C^\infty$. Results. We give an estimate from above of groups of motions of the tangent bundle $TM_2$, equipped with a complete mobile lift maximum-affine connection with a non-zero curvature tensor field. In the same section we are building Lie algebra of infinitesimal automorphisms of spaces ($TM_2,\bigtriangledown ^{(0)}$) over a manifold $M_2$ of connectivity with the appropriate components, and solve the problem of solvability of this algebra. Conclusions. The Lie algebra $\widetilde{L}$ of infinitesimal affine transformations of the space ($TM_2,\bigtriangledown ^{(0)}$) above the maximum-moving two-dimensional space ($M_2,\bigtriangledown $), defined by the connection coefficients (6), is solvable.

Keywords: affine transformations, Lie algebra, tangent bundles, automorphism.

UDC: 514.76

DOI: 10.21685/2072-3040-2016-4-4