Article 5114

Title of the article

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

Authors

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

Index UDK

514.76

Abstract

Background. Vector fields of type γG, GHγ,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 -γr 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 , 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 γr -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 γr -lift as a mapping γr ℑr1(M) →ℑ01 (T (M)) , which in local coordinates ( xi0, xi1) determined by the condition γrG=Gij1...jrx1j1...x1jr1j, where ∂i = ∂/∂xi.

Key words

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

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References

1. Yano K. and Ishihara S. Marcel Dek-ker, Inc. New York, 1973, pp. 12–25.
2. Sultanov A. Ya. Trudy geometricheskogo seminara [Proceedings of the seminar on geometry]. Issue 22. Kazan': Izd-vo Kazan. un-ta, 1994, pp. 78–88.
3. Shadyev X. Trudy geometricheskogo seminara [Proceedings of the seminar on geometry]. Issue 16. Kazan': Izd-vo Kazan. un-ta, 1984, pp. 117–127.

 

Дата создания: 18.07.2014 13:27
Дата обновления: 23.07.2014 15:07