ON LIE ALGEBRAS OF INFINITESIMAL AFFINE TRANSFORMATIONS IN TANGENT
BUNDLES WITH A COMPLETE LIFT CONNECTION  

Sultanova Galiya Alievna, Postgraduate student, Penza State University (40 Krasnaya street, Penza, Russia), sultgaliya@yandex.ru

514.76

10.21685/2072-3040-2016-4-4

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 infini tesimal 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 twodimensional 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  (TM2 (V0)) and clarified the question of solvability of this algebra.
Materials and methods. The object of study is the space (TM2 (V0)) . 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 infinity .
Results. We give an estimate from above of groups of motions of the tangent bundle TM2 , 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 (TM2 (V0)) over a manifold M2 of connectivity with the appropriate components, and solve the problem of solvability of this algebra.
Conclusions. The Lie algebra L  of infinitesimal affine transformations of the space (TM2 (V0)) above the maximum-moving two-dimensional space  M2V , defined by the connection coefficients (6), is solvable.  

affine transformations, Lie algebra, tangent bundles, automorphism

1. Yano K., Ishihara S. Tangent and cotangent bundles. Differential Geometry. New York, Marcel Dekker, 1973, 423 p.
2. Shadyev Kh. Trudy geometricheskogo seminara [Proceedings of a geometrical seminar]. 1984, vol. 16, pp. 117–127.
3. Egorov I. P. Uchenye zapiski Penzenskogo pedagogicheskogo instituta [Proceedings of Penza Pedagogical Institute]. Kazan: Izd-vo Kazanskogo gos. un-ta, 1965, pp. 5–179.
4. Tanno S. Infinitesimal isometries on the tangent bundles with complete lift metric. Tensor, N.S. 1974, vol. 28, pp. 139–144.
5. Sultanov A. Ya. Izvestiya vuzov. Matematika [University proceedings. Mathematics]. 1999, no. 9, pp. 64–72.
6. Sato J. Kodai Math. Semin. Repts. 1968, vol. 20, no. 4, pp. 458–468.
7. Sultanova G. A. Differentsial'naya geometriya mnogoobraziy figur [Differential geometry of figure manifold]. Issue. 46. Kaliningrad, 2015, pp. 153–161.
8. Pontryagin L. S. Nepreryvnye gruppy: monogr. [Continuous groups: monograph]. Moscow: Nauka, Fizmatlit, 1973, 527 p.