On the Lifts of $F^{K}+F=0,$ $(F\neq 0,K\geqslant 0)-$Structure on Cotangent and Tangent Bundle

Abstract

This paper consists of two main sections. In the first part, we find the integrability conditions by calculating Nijenhuis tensors of the horizontal lifts of $F(K,1)-$structure satisfying $F^{K}+F=0$. Later, we get the results of Tachibana operators applied to vector and covector fields according to the horizontal lifts of $F(K,1)-$structure in cotangent bundle $ T^{\ast }(M^{n})$. Finally, we have studied the purity conditions of Sasakian metric with respect to the horizontal lifts of $F(K,1)-$structure. In the second part, all results obtained in the first section were obtained according to the complete and horizontal lifts of $F(K,1)-$structure in tangent bundle $T(M^{n})$.

Keywords

• Integrability conditions
• Tachibana operators
• lifts
• Sasakian metric
• tangent bundle
• cotangent bundle

References

1. [1] Andreou, F.G., On integrability conditions of a structure f satisfying f⁵+f=0, Tensor, N.S., 40: 27-31, (1983).
2. [2] Çayır, H., Some Notes on Lifts of Almost Paracontact Structures, American Review of Mathematics and Statistics, 3(1): 52-60, (2015).
3. [3] Çayır, H., Lie derivatives of almost contact structure and almost paracontact structure with respect to X^{V} and X^{H} on tangent bundle T(M), Proceedings of the Institute of Mathematics and Mechanics, 42(1): 38-49, (2016).
4. [4] Çayır, H., Tachibana and Vishnevskii Operators Applied to X^{V} and X^{H} in Almost Paracontact Structure on Tangent Bundle T(M), New Trends in Mathematical Sciences, 4(3): 105-115, (2016).
5. [5] Çayır, H. and Köseoğlu, G., Lie Derivatives of Almost Contact Structure and Almost Paracontact Structure With Respect to X^{C} and X^{V} on Tangent Bundle T(M), New Trends in Mathematical Sciences, 4(1): 153-159, (2016).
6. [6] Gupta, V.C., Integrability Conditions of a Structure F Satisfying F^{K}+F=0, The Nepali Math. Sc. Report, 14: 55-62, (1998).
7. [7] Ishıhara, S. and Yano, K., On integrability conditions of a structure f satisfying f³+f=0, Quaterly J. Math., 15: 217-222, (1964).
8. [8] Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry-Volume I. John Wiley & Sons, Inc, New York, (1963).

9. [9] Lovejoy, S.D., Nivas, R. and Pathak, V.N., On horizontal and complete lifts from a manifold with fλ(7,1)-structure to its cotangent bundle, International Journal of Mathematics and Mathematical Sciences, 8: 1291-1297, (2005).
10. [10] Nivas, R. and Prasad, C.S., On a structure defined by a tensor field f(≠0) of type (1,1) satisfying f⁵-a²f=0. Nep. Math. Sc. Rep., 10(1): 25-30, (1985).
11. [11] Salimov, A.A., Tensor Operators and Their applications, Nova Science Publ., New York, (2013).
12. [12] Salimov, A.A. and Çayır, H., Some Notes On Almost Paracontact Structures, Comptes Rendus de 1'Acedemie Bulgare Des Sciences, 66(3): 331-338, (2013).
13. [13] Yano, K. and Patterson, E.M., Horizontal lifts from a manifold to its cotangent bundle, J. Math. Soc. Japan 19: 185-198, (1967).
14. [14] Yano, K. and Ishihara, S., Tangent and Cotangent Bundles, Marcel Dekker Inc., New York, (1973).