In this paper, M is a smooth manifold of finite dimension n, A is a local algebra and MA is the associated Weil bundle. We study Poisson vector fields on MA and we prove that all globally hamiltonian vector fields on MA are Poisson vector fields.
References
[1]
Weil, A. (1953) Théorie des points proches sur les variétés différentiables. Colloq. Géom. Diff. Strasbourg, 111-117.
[2]
Morimoto, A. (1976) Prolongation of Connections to Bundles of Infinitely Near Points. Journal of Differential Geometry, 11, 479-498.
[3]
Okassa, E. (1986-1987) Prolongement des champs de vecteurs à des variétés des points prohes. Annales de la Faculté des Sciences de Toulouse, 3, 346-366.
[4]
Bossoto, B.G.R. and Okassa, E. (2008) Champs de vecteurs et formes différentielles sur une variété des points proches. Archivum Mathematicum, Tomus, 44, 159-171.
[5]
Kolár, P., Michor, P.W. and Slovak, J. (1993) Natural Operations in Differential Geometry. Springer-Verlag, Berlin. http://dx.doi.org/10.1007/978-3-662-02950-3
[6]
Moukala Mahoungou, N. and Bossoto, B.G.R. (2015) Hamiltonian Vector Fields on Weil Bundles. Journal of Mathematics Research, 7, 141-148. http://dx.doi.org/10.5539/jmr.v7n3p141
[7]
Laurent-Gengoux, C., Pichereau, A. and Vanhaecke, P. (2013) Poisson Structures. Grundlehren der mathematischen Wissenschaften, 347. www.springer.com/series/138
[8]
Okassa, E. (2007) Algèbres de Jacobi et Algèbres de Lie-Rinehart-Jacobi. Journal of Pure and Applied Algebra, 208, 1071-1089. http://dx.doi.org/10.1016/j.jpaa.2006.05.013
[9]
Moukala Mahoungou, N. and Bossoto, B. G.R. Prolongation of Poisson 2-Form on Weil Bundles.
[10]
Lichnerowicz, A. (1977) Les variétés de Poisson et leurs algèbres de Lie associées. Journal of Differential Geometry, 12, 253-300.
[11]
Vaisman, I. (1994) Lectures on the Geometry of Poisson Manifolds. Progress in Mathematics 118, Birkhäuser Verlag, Basel. http://dx.doi.org/10.1007/978-3-0348-8495-2
[12]
Bossoto, B.G.R. and Okassa, E. (2012) A-Poisson Structures on Weil Bundles. International Journal of Contemporary Mathematical Sciences, 7, 785-803.
[13]
Nkou, V.B., Bossoto, B.G.R. and Okassa, E. (2015) New Characterization of Vector Field on Weil Bundles. Theoretical Mathematics and Applications, 5, 1-17.
