DERIVATIVES WITH RESPECT TO HORIZONTAL AND VERTICAL LIFTS OF THE CHEEGER-GROMOLL METRIC $^{CG}g$ ON THE $(1,1)-$TENSOR BUNDLE $T_{1}^{1}(M)$

Yıl 2017, Cilt: 5 Sayı: 2, 78 - 86, 15.10.2017

HAŞİM Çayır MOHAMMAD NAZRUL ISLAM Khan

Öz

In this paper, we define the Cheeger-Gromoll metric in the $(1,1)$ $-$tensor bundle $T_{1}^{1}(M)$, which is completely determined by its action on vector fields of type $X^{H}$ and $\omega ^{V}$. Later, we obtain the covarient and Lie derivatives applied to the Cheeger-Gromoll metric with respect to the horizontal and vertical lifts of vector and kovector fields, respectively.

Anahtar Kelimeler

(1 1)-tensor bundle, Covarient Derivative, Lie Derivative, Cheeger-Gromoll metric, Horizontal Lift, Vertical Lift

Kaynakça

• [1] Akyol, M. A., Sarı, R. and Aksoy, E., Semi-invariant -Riemannian submersions from almost contact metric manifolds, Int. J. Geom. Methods Mod. Phys. 14, 175007 4 (2017) DOI:http://dx.doi.org/10.1142/S0219887817500748.
• [2] Akyol, M. A., Conformal anti-invariant submersions from cosymplectic manifolds, Hacet. J. Math. Stat. 46(2017), no.2, 177-192.
• [3] Çakmak, A. and Tarakç, Ö., Surfaces at a constant distance from the edge of regression on a surface of revolution in . Applied Mathematical Sciences, 10(2016), no.15, 707-719.
• [4] Çakmak, A., Karacan, M.K., Kiziltug, S. and Yoon, D.W., Translation surfaces in the 3-dimensional Gallean space satisfying . Bull. Korean Math. Soc. https://doi.org/10.4134/BKMS.b160442.
• [5] Çayır, H. and Akdağ, K., Some notes on almost paracomplex structures associated with the diagonal lifts and operators on cotangent bundle, New Trends in Mathematical Sciences, 4(2016), no.4, 42-50.
• [6] Çayır, H. and Köseoğlu, 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.
• [7] Cengiz, N. and Salimov, A. A., Complete lifts of derivations to tensor bundles, Bol. Soc. Mat. Mexicana (3) 8(2002), no.1, 75-82.
• [8] Gancarzewicz, J. and Rahmani, N., Relevent horizontal des connexions linearies au bre vectoriel associe avec le bre principal des repres lineaires, Annales Polinici Math., 48(1988), 281-289.
• [9] Gezer, A. and Altunbas, M., On the (1; 1)-tensor bundle with Cheeger-Grommol type metric, Proc. Indian Acad. Sci.(Math Sci.) 125(2015), no.4, 569-576.
• [10] Gunduzalp, Y., Slant submersions from almost paracontact Riemannian manifolds, product Riemannian manifolds, Kuwait Journal of Science, 42(2015), no.1, 17-29.
• [11] Gunduzalp, Y., Semi-slant submersions from almost product Riemannian manifolds, DEMONSTRATIO MATHEMATICA, 49(2016), no.4.
• [12] Khan, M. N. I., and Jun, J.B., Lorentzian Almost r-para-contact Structure in Tangent Bundle, Journal of the Chungcheong Mathematical Society, 27(2014), no.1, 29-34.
• [13] Kobayashi, S. and Nomizu, K., Foundations of Dierential Geometry-Volume I, John Wiley & Sons, Inc, New York, 1963.
• [14] Lai, K. F. and Mok, K. P., On the differential geometry of the (1; 1)-tensor bundle, Tensor (New Series), 63(2002), no.1, 15-27.
• [15] Ledger, A. J. and Yano, K., Almost complex structures on the tensor bundles, J. Diff. Geom., 1(1967), 355-368.
• [16] Salimov, A.A., Tensor Operators and Their applications, Nova Science Publ., New York, 2013.
• [17] Salimov, A. and Gezer, A., On the geometry of the (1,1) -tensor bundle with Sasaki type metric, Chin. Ann. Math. Ser. B 32(2011), no.3, 369-386.
• [18] Yano, K. and Ishihara, S., Tangent and Cotangent Bundles, Marcel Dekker, New York, 1973.