Fıdan JABRAILZADE, Haşim ÇAYIR

On the Lifts of F(2K + S; S)?Structure Satisfying F2K+S + FS = 0; (F 6= 0; K 1 1; S 1 1) on Cotangent and Tangent Bundle

This paper consists of two main sections. In the first part, we find the integrability conditions bycalculating Nijenhuis tensors of the horizontal lifts of F(2K + S; S)?structure Satisfying F2K+S +FS = 0. Later, we get the results of Tachibana operators applied to vector and covector fieldsaccording to the horizontal lifts of F(2K + S; S)?structure in cotangent bundle T(Mn). Finally,we have studied the purity conditions of Sasakian metric with respect to the horizontal lifts of thestructure. In the second part, all results obtained in the first section were obtained according to thecomplete and horizontal lifts of F(2K + S; S)?structure in tangent bundle T(Mn).

PDF

___

[1] Akyol, M. A. and Gündüzalp, Y., Hemi-Slant Submersions from Almost Product Riemannian Manifolds. Gulf Journal of Mathematics 4 (2016), no. 3, 15-27.
[2] Akyol, M. A. and Gündüzalp, Y., Hemi-Invariant Semi-Riemannian Submersions. Commun. Fac. Sci. Univ. Ank. Series A1 67 (2018), no. 1, 80-92.
[3] Andreou, F. G., On integrability conditions of a structure f satisfying f5 + f = 0. Tensor, N.S. 40 (1983), 27-31.
[4] Cengiz, N. and Salimov, A. A., Diagonal Lift in the Tensor Bundle and its Applications. Applied Mathematics and Computation 142 (2003) 309-319.
[5] Çayır, H., Some Notes on Lifts of Almost Paracontact Structures. American Review of Mathematics and Statistics 3 (2015), no. 1, 52-60.
[6] Çayır, H., Lie derivatives of almost contact structure and almost paracontact structure with respect to XV and XH on tangent bundle T(M). Proceedings of the Institute of Mathematics and Mechanics 42 (2016), no. 1, 38-49.
[7] Çayır, H., Tachibana and Vishnevskii Operators Applied to XV and XH in Almost Paracontact Structure on Tangent Bundle T(M). New Trends in Mathematical Sciences 4 (2016), no. 3, 105-115.
[8] Çayır, H. and Köseo˘ glu, G., Lie Derivatives of Almost Contact Structure and Almost Paracontact Structure With Respect to XC and XV on Tangent Bundle T(M). New Trends in Mathematical Sciences 4 (2016), no. 1, 153-159.
[9] Das, L., On CR-structure and an f(2K + 4; 2)?structure satisfying f2K+4 + f2 = 0. Tensor 73 (2011), no. 3, 222-227.
[10] Das, L., On lifts of structure satisfying FK+1 ? a2FK?1 = 0. Kyungpook Mathematical Journal 40 (2000), no. 2, 391-398.
[11] Gupta, V.C., Integrability Conditions of a Structure F Satisfying FK + F = 0: The Nepali Math. Sc. Report 14 (1998), no. 2, 55-62.
[12] Ishıhara, S. and Yano, K., On integrability conditions of a structure f satisfying f3 + f = 0: Quaterly J. Math. 15 (1964), 217-222.
[13] Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry-Volume I. John Wiley & Sons, Inc, New York, 1963.
[14] Lovejoy, S. D., Nivas, R. and Pathak, V. N., On horizontal and complete lifts from a manifold with f(7; 1)?structure to its cotangent bundle. International Journal of Mathematics and Mathematical Sciences 8 (2005), 1291-1297.
[15] Musso, E. and Tricerri, F., Riemannian Metric on Tangent Bundles. Ann. Math. Pura. Appl. 150 (1988), no. 4, 1-9.
[16] Nivas, R. and Prasad, C. S., On a structure defined by a tensor field f(6= 0) of type (1; 1) satisfying f5 ? a2f = 0. Nep. Math. Sc. Rep. 10 (1985), no. 1, 25-30.
[17] Sahin, B.,Anti-invariant Riemannian submersions from almost Hermitian manifolds. CentralEuropean J.Math. 3 (2010), 437-447.
[18] Sahin, B., Semi-invariant Riemannian submersions from almost Hermitian manifolds. Canad.Math. Bull. 56 (2013), 173-183.
[19] Salimov, A. A., Tensor Operators and Their applications. Nova Science Publ., New York, 2013.
[20] Salimov, A. A. and Çayır, H., Some Notes On Almost Paracontact Structures. C. R. Acad. Bulgare Sci. 66 (2013), no. 3, 331-338.
[21] Singh, A., On CR?structures F?structures satisfying F2K+P + FP = 0. Int. J. Contemp. Math. Sciences 4 (2009), 1029-1035.
[22] Singh, A., Pandey, R. K. and Khare, S., On horizontal and complete lifts of (1; 1) tensor fields F satisfying the structure equation F(2K + S; S) = 0. International Journal of Mathematics and Soft Computing 6 (2016), no. 1, 143-152.
[23] Soylu, Y., A Myers-type compactness theorem by the use of Bakry-Emery Ricci tensor. Differ. Geom. Appl. 54 (2017), 245–250.
[24] Soylu, Y., A compactness theorem in Riemannian manifolds. J. Geom. 109 (2018) no. 1, Art. 20, 6 pp.
[25] Yano, K. and Patterson, E. M., Horizontal lifts from a manifold to its cotangent bundle. J. Math. Soc. Japan 19 (1967), 185-198.
[26] Yano, K. and Ishihara, S., Tangent and Cotangent Bundles. Marcel Dekker Inc., New York, 1973.