SOME LIFTS OF TENSOR FIELDS OF TYPE (1, R) WITH BASE IN ITS TANGENT BUNDLE

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

514.76

Background. Vector fields of type γG, G^Hγ,representing lifts of tensor field G∈ℑ₁¹(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 -γ^r lifts of tensor fields of type (1, r), (r ≥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 T(M) 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∞ class.
Results. In paragraph 2 of this paper there were discovered commutators of vector fields In paragraph 2 of this paper there were discovered commutators of vector fields  γG , G^Hγ , where G∈ℑ₁¹(M) , and there was introduced a definition of γγ -lift of tensor field of type (1,2). In paragraph 3 there were proved some of properties of γγ -lift. In paragraph 4 there was constructed γ^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 γ^r -lift as a mapping γ^r ℑr¹(M) →ℑ₀¹ (T (M)) , which in local coordinates ( xⁱ₀, xⁱ₁) determined by the condition γ^rG=Gⁱ_j1...jrx_1^j1...x_1^jr∂¹_j, where ∂_i = ∂/∂x^i.

smooth manifold, tangent bundle, lifts of tensor fields, tensor field, commutator of vector fields.

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