SOME LIFTS OF TENSOR FIELDS OF TYPE (1, <I>R</I>) WITH BASE IN ITS TANGENT BUNDLE
Background. Vector fields of type, representing lifts of tensor field, defined on a smooth manifold M to the tangent bundle, arise in the study of infinitesimal affine transformations with a complete lift. These lifts were introduced by K.Yano, Sh.Ishihara and used by F.I. Kagan in the study of infinitesimal affine, infinitesimal projective transformations equipped with full lift of torsion-free linear connection, defined on the basis of M, used by H.Shadyev when he described infinitesimal affine transformations of synectic lift of linear torsion-free connection with a smooth manifold M to its tangent bundle. The purpose of this paper is to construct -lifts of tensor fields of type (1, r ), and explain some of their properties with respect to the differentiation of Lee and covariant differentiation. Materials and methods. The object of the study is the tangent bundle of a smooth manifold M. There are used the methods of tensor analysis, the theory of the Lie derivative. The manifold, functions, tensor fields were assumed to be the smooth of class. Results. In paragraph 2 of this paper there were discovered commutators of vector fields, where, and there was introduced a definition of -lift of tensor field of type (1,2). In paragraph 3 there were proved some of properties of -lift. In paragraph 4 there was constructed -lift for any tensor field of type (1, r ) and the properties were proved. Conclusions. For any tensor field of type (1, r ) it is possible to construct -lift as a mapping, which in local coordinates is determined by the condition, where.