Vertical and complete lifts of sections of a (dual) vector bundle and Legendre duality

Year 2019, Volume: 68 Issue: 1, 299 - 325, 01.02.2019

E. Peyghan , L. Nourmohammadifar C. M. Arcuş

https://doi.org/10.31801/cfsuasmas.415924

Abstract

Using the covariant derivative for exterior forms of a (dual) vector bundle, the complete lift of an arbitrary section of a (dual) vector bundle is discovered. A theory of Legendre type and Legendre duality between vertical lifts and between complete lifts are presented. Finally, a duality between Lie algebroids structures is developed

Keywords

Vertical and complete lifts of sections of a (dual) vector bundle and Legendre duality

References

• Arcuş, C. M., Generalized Lie algebroids and connections over pair of diffeomorphic manifolds, J. Gen. Lie Theory Appl. 7, (2013), 32 pages. Arcuş, C. M., Mechanical systems in the generalized Lie algebroids framework, Int. J. Geom. Methods Mod. Phys. 11, (2014), 40 pages. Arcuş, C. M. and Peyghan, E., (Pseudo) generalized Kaluza--Klein G-spaces and Einstein equations, International Journal of Mathematics 25(12), (2014), doi: 10.1142/S0129167X1450116X.
• Arcuş, C.M., Peyghan, E. and Sharahi, E., Weyl's theory in the generalized Lie algebroids framework, J. Math Phys. 55, (2014), doi: 10.1063/1.4903256.
• Esin, E. and Civelek, S., The lifts on the second order tangent bundles, J. Math. Stat. Fac. Art. Sc. Gazi Univ. 2, (1989), 117-135.
• de León M., Marrero J.C., Martínez E.: Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen. 38 (2005), 241-308.
• Martínez, E., Lagrangian mechanics on Lie algebroids, Acta Appl. Math. 67, (2001), 295-320. OSH : Omran, T., Sharffuddin, A. and Husain, S.I., Lifts of structures on manifolds, Publications De L'institut Math. 36(50), (1984), 93-97.
• Özkan, M., Prolongations of golden structures to tangent bundles, Differential Geometry-Dynamical Systems 16, (2014), 227-238.
• Özkan, M., Çıtlak, A. A. and Taylan E., Prolongations of golden structure to tangent bundle of Order 2, Gazi University Journal of Science 28(2), (2015), 253-258.
• Özkan, M. and Yılmaz F., Prolongations of golden structures to tangent bundles of order r, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 65(1), (2016), 35-47.
• Özkan, M., Taylan, E. and Çıtlak, A. A., On lifts of silver structure, Journal of Science and Arts 2(39), (2017), 223-234.
• Peyghan, E., Models of Finsler Geometry on Lie algebroids, arXiv: 1310.7393v1, (2013), 90 pages.
• Peyghan, E., Nasrabadi, H. and Tayebi, A., The homogenous lift of the (1, 1)-tensor bundle of a Riemannian metric, Int. J. Geom. Methods Mod. Phys. 10, (2013), 18 pages.
• Popescu, L., The geometry of Lie algebroids and its applications to optimal control, arXiv:1302.5212v2 [Math.DG] 25 Feb 2013.
• Popescu, L., A note on Poisson-Lie algebroids, J. Geom. Symmetry Phys. 12, (2008), 63-73. Salimov, A.A and Magden, A., Complete lifts of tensor fields on a pure cross-section in the tensor bundle T_{q}¹(M_{n}), Note di Matematica 18, (1998), 27-37.
• Sarlet, W. and Waeyaert, G., Lifting geometric objects to the dual of the first jet bundle of a bundle fibred over R, J. Geom and Phys. 74, (2013), 109-118.
• Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. (I, 10 (1958) 338-354; II, 14 (1962) 146-155).
• Yano, K. and Ishihara, S., Tangent and Cotangent Bundles, Marcel Dekker, 1973.