Previous |  Up |  Next

Article

Keywords:
Weil bundle; Weil algebra; Poisson manifold; Lie derivative; Poisson 2-form
Summary:
In this paper, $M$ denotes a smooth manifold of dimension $n$, $A$ a Weil algebra and $M^{A}$ the associated Weil bundle. When $(M,\omega _{M})$ is a Poisson manifold with $2$-form $\omega _{M}$, we construct the $2$-Poisson form $\omega _{M^{A}}^{A}$, prolongation on $M^{A}$ of the $2$-Poisson form $\omega _{M}$. We give a necessary and sufficient condition for that $M^{A}$ be an $A$-Poisson manifold.
References:
[1] Bossoto, B.G.R., Okassa, E.: Champs de vecteurs et formes différentielles sur une variété des points proches. Arch. Math. (Brno) 44 (2008), 159–171. MR 2432853 | Zbl 1212.13016
[2] Bossoto, B.G.R., Okassa, E.: A-poisson structures on Weil bundles. Int. J. Contemp. Math. Sci. 7 (16) (2012), 785–803. MR 2901677 | Zbl 1247.53093
[3] Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin, 1993. MR 1202431
[4] Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P.: Poisson Structures. Grundlehren Math. Wiss. 347 (2013), www.springer.com/series/138. MR 2906391
[5] Lichnerowicz, A.: Les variétés de Poisson et leurs algèbres de Lie associées. J. Differential Geom. 12 (1977), 253–300. MR 0501133 | Zbl 0405.53024
[6] Morimoto, A.: Prolongation of connections to bundles of infinitely near points. J. Differential Geom. 11 (1976), 479–498. MR 0445422 | Zbl 0358.53013
[7] Moukala, N.M., Bossoto, B.G.R.: Hamiltonian vector fields on Weil bundles. Journal of Mathematics Research 7 (3) (2015), 141–148. DOI 10.5539/jmr.v7n3p141
[8] Nkou, V.B., Bossoto, B.G.R., Okassa, E.: New characterization of vector field on Weil bundles. Theoretical Mathematics $\&$ Applications 5 (2) (2015), 1–17, arXiv:1504.04483 [math.DG].
[9] Okassa, E.: Prolongement des champs de vecteurs à des variétés des points proches. Ann. Fac. Sci. Toulouse Math. (5) 8 (3) (1986–1987), 346–366. MR 0948759
[10] Okassa, E.: Algèbres de Jacobi et algèbres de Lie-Rinehart-Jacobi. J. Pure Appl. Algebra 208 (3) (2007), 1071–1089. DOI 10.1016/j.jpaa.2006.05.013 | Zbl 1163.17025
[11] Okassa, E.: On Lie-Rinehart-Jacobi algebras. J. Algebra Appl. 7 (2008), 749–772. DOI 10.1142/S0219498808003107 | MR 2483330 | Zbl 1226.17017
[12] Okassa, E.: Symplectic Lie-Rinehart-Jacobi algebras and contact manifolds. Canad. Math. Bull. 54 (4) (2011), 716–725. DOI 10.4153/CMB-2011-033-6 | MR 2894521 | Zbl 1232.53062
[13] Shurygin, V.V.: Some aspects of the theory of manifolds over algebras and of Weil bundles. J. Math. Sci. (New York) 169 (3) (2010), 315–341. DOI 10.1007/s10958-010-0051-6 | MR 2866746
[14] Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Progress in Math., vol. 118, Birkhäuser Verlag, Basel, 1994. MR 1269545 | Zbl 0810.53019
[15] Weil, A.: Théorie des points proches sur les variétés différentiables. Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg (1953), 111–117. MR 0061455 | Zbl 0053.24903
Partner of
EuDML logo