Some lifts of tensor fields of type $(1, r)$ with base in its tangent bundle

Background. Vector fields of type $\gamma G, G^{H\gamma}$ representing lifts of tensor field $G \in \Im^1_1(M) $, defined on a smooth manifold M to the tangent bundle T(M), 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 T(M). The purpose of this paper is to construct $\gamma ^r$-lifts of tensor fields of type $(1, r)$, $(r\geq 1)$ 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 T9m) 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 $C^\infty$ class. Results. In paragraph 2 of this paper there were discovered commutators of vector fields $\gamma G, G^{H\gamma}$ where $G \in \Im^1_1(M) $, and there was introduced a definition of $\gamma \gamma$-lift of tensor field of type (1,2). In paragraph 3 there were proved some of properties of $\gamma^r$-lift. In paragraph 4 there was constructed $\gamma^r$-lift for any tensor field G of type $(1, r)$ and the properties were proved. Conclusions. For any tensor field G of type $(1, r)$ it is possible to construct $\gamma^r$-lift as a mapping $\gamma ^r : \Im^1_r(M) \rightarrow \Im^1_0(T(M))$, which in local coordinates $(x^i_0,x^i_1)$ is determined by the condition $\gamma ^r G=G^i _{j_1,...,j_r} x^{j_1}_1 ... x^{j_r}_1 \partial^1_i$, where $\partial_i = \frac{\partial}{\partial x^i} $.