Abstract
Let Q be a smoothmanifold of dimension n ≥ 1. In this paper,we define the vertical lift of multivector fields from Q to T Q and we give some applications in the Poisson geometry. In particular we describe the structure of singular foliation induced by the vertical lift of Poisson structures defined below. On the other hand, given a second order vector field on Q, we defined the horizontal lift of multivector fields from Q to TQ and we study some properties.
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References
T. Courant, “Tangent Lie algebroids,” J. Phys. A: Math. Gen. 23, 4527–4536 (1994).
T. Courant, “Tangent Dirac structures,” J. Phys. A: Math.Gen. 23, 5153–5168 (1990).
J. Grabowski and P. Urbanski, “Tangent lifts of Poisson and related structures,” J. Phys. A: Math. Gen. 28, 6743–6777 (1995).
I. Kolar, P. Michor, and J. Slovak, Natural Operations in Differential Geometry (Springer, Berlin, 1993).
Y. Kosmann-Schwarzbach, “The Lie bialgebroid of a Poisson–Nijenhuis manifold,” Lett. Math. Phys. 38, 421–428 (1996).
Y. Kosmann-Schwarzbach and F. Magri, “On the modular class of Poisson–Nijenhuis manifolds,” arXiv: math/0611202v1 [math.SG] (November 7, 2006).
Y. Kosmann-Schwarzbach and F. Magri, “Poisson–Nijenhuis structures,” Ann. Inst. Henri Poincare 53, 35–81 (1990).
P. M. Kouotchop Wamba and A. J. Ntyamand Wouafo Kamga, “Tangent lifts of higher order of multivector fields and applications,” J. Math. Sci.: Adv. Appl. 15 (2), 89–112 (2012).
K. Mackenzie and P. Xu, “Lie bialgebroids and Poisson groupoids,” Duke Math. J. 73, 415–452 (1998).
G. Mitric and I. Vaisman, “Poisson structures on tangent bundles,” Diff. Geom. Appl. 18, 207–228 (2003).
A. Morimoto, “Lifting of some type of tensors fields and connections to tangent bundles of pr-velocities,” Nagoya Math. 40, 13–31 (1970).
T. Vaisman, “The Poisson–Nijenhuis manifolds revisited,” Rend. Sem. Mat. Univ. Poi. Torino 52 (4) (1994).
J. Wouafo Kamga, On the Tangential Linearization of Hamiltonian Systems (Int. Centre for Theor. Phys., Trieste, Italy, 1997).
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Submitted by M. A.Malakhaltsev
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Mba, A., Wamba, P.M.K. & Nimpa, R.P. Vertical and horizontal lifts of multivector fields and applications. Lobachevskii J Math 38, 1–15 (2017). https://doi.org/10.1134/S1995080217010140
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DOI: https://doi.org/10.1134/S1995080217010140