Note on the cross-section in the semi-tensor bundle

Abstract

Using the fiber bundle M over a manifold B, we define a semi-tensor (pull-back) bundle tB of type (p,q). The complete and horizontal lift of projectable geometric objects on M to the semi-tensor (pull-back) bundle tB of type (p,q) are presented. The main purpose of this paper is to study the behaviour of complete lift of vector and affinor (tensor of type (1,1)) fields on cross-sections for pull-back (semi-tensor) bundle tB of type (p,q).

Keywords

• Vector field,complete lift,horizontal lift,pull-back bundle,cross-section

References

1. T.V. Duc, Structure presque-transverse. J. Diff . Geom., 14(1979), No:2, 215-219.
2. C.J. Isham, ”Modern differential geometry for physicists”, World Scientific, 1999.
3. H. Fattaev, The Lifts of Vector Fields to the Semitensor Bundle of the Type (2, 0), Journal of Qafqaz University, 25 (2009), no. 1, 136-140.
4. A. Gezer and A. A. Salimov, Almost complex structures on the tensor bundles, Arab. J. Sci. Eng. Sect. A Sci. 33 (2008), no. 2, 283–296.
5. D. Husemoller, Fibre Bundles. Springer, New York, 1994.
6. V. Ivancevic and T. Ivancevic , Applied Differential Geometry, A Modern Introduction, World Scientific, Singapore, 2007.
7. H.B. Lawson and M.L. Michelsohn, Spin Geometry. Princeton University Press., Princeton, 1989.
8. A.J. Ledger and K. Yano, Almost complex structure on tensor bundles, J. Dif. Geom. 1 (1967), 355-368.

9. A. Salimov, Tensor Operators and their Applications. Nova Science Publ., New York, 2013.
10. A. A. Salimov and E. Kadıo˘glu, Lifts of derivations to the semitangent bundle, Turk J. Math. 24 (2000), 259-266.
11. N. Steenrod, The Topology of Fibre Bundles. Princeton University Press., Princeton, 1951.
12. V. V. Vishnevskii, Integrable affinor structures and their plural interpretations. Geometry, 7.J. Math. Sci. (New York) 108 (2002), no. 2, 151-187.
13. G. Walschap, Metric Structures in Differential Geometry, Graduate Texts in Mathematics, Springer-Verlag, New York, 2004.
14. F. Yıldırım, On a special class of semi-cotangent bundle, Proceedings of the Institute of Mathematics and Mechanics, (ANAS) 41 (2015), no. 1, 25-38.
15. F. Yıldırım and A. Salimov, Semi-cotangent bundle and problems of lifts, Turk J. Math, (2014), 38, 325-339.