Geodesic
Prolongation of Poisson $2$-form on Weil bundles
Archivum mathematicum, Tome 52 (2016) no. 2, pp. 91-111 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.5817/AM2016-2-91
Classification : 17D63, 53D05, 53D17, 58A20, 58A32
Keywords: Weil bundle; Weil algebra; Poisson manifold; Lie derivative; Poisson 2-form 
@article{10_5817_AM2016_2_91,
     author = {Moukala, Norbert Mahoungou and Bossoto, Basile Guy Richard},
     title = {Prolongation of {Poisson} $2$-form on {Weil} bundles},
     journal = {Archivum mathematicum},
     pages = {91--111},
     year = {2016},
     volume = {52},
     number = {2},
     doi = {10.5817/AM2016-2-91},
     mrnumber = {3535631},
     zbl = {06644061},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2016-2-91/}
}
TY  - JOUR
AU  - Moukala, Norbert Mahoungou
AU  - Bossoto, Basile Guy Richard
TI  - Prolongation of Poisson $2$-form on Weil bundles
JO  - Archivum mathematicum
PY  - 2016
SP  - 91
EP  - 111
VL  - 52
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2016-2-91/
DO  - 10.5817/AM2016-2-91
LA  - en
ID  - 10_5817_AM2016_2_91
ER  - 
%0 Journal Article
%A Moukala, Norbert Mahoungou
%A Bossoto, Basile Guy Richard
%T Prolongation of Poisson $2$-form on Weil bundles
%J Archivum mathematicum
%D 2016
%P 91-111
%V 52
%N 2
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2016-2-91/
%R 10.5817/AM2016-2-91
%G en
%F 10_5817_AM2016_2_91
Moukala, Norbert Mahoungou; Bossoto, Basile Guy Richard. Prolongation of Poisson $2$-form on Weil bundles. Archivum mathematicum, Tome 52 (2016) no. 2, pp. 91-111. doi: 10.5817/AM2016-2-91

[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 | Zbl

[2] Bossoto, B.G.R., Okassa, E.: A-poisson structures on Weil bundles. Int. J. Contemp. Math. Sci. 7 (16) (2012), 785–803. | MR | Zbl

[3] Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin, 1993. | MR

[4] Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P.: Poisson Structures. Grundlehren Math. Wiss. 347 (2013), www.springer.com/series/138. | MR

[5] Lichnerowicz, A.: Les variétés de Poisson et leurs algèbres de Lie associées. J. Differential Geom. 12 (1977), 253–300. | DOI | MR | Zbl

[6] Morimoto, A.: Prolongation of connections to bundles of infinitely near points. J. Differential Geom. 11 (1976), 479–498. | DOI | MR | Zbl

[7] Moukala, N.M., Bossoto, B.G.R.: Hamiltonian vector fields on Weil bundles. Journal of Mathematics Research 7 (3) (2015), 141–148. | DOI

[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

[10] Okassa, E.: Algèbres de Jacobi et algèbres de Lie-Rinehart-Jacobi. J. Pure Appl. Algebra 208 (3) (2007), 1071–1089. | DOI | MR | Zbl

[11] Okassa, E.: On Lie-Rinehart-Jacobi algebras. J. Algebra Appl. 7 (2008), 749–772. | DOI | MR | Zbl

[12] Okassa, E.: Symplectic Lie-Rinehart-Jacobi algebras and contact manifolds. Canad. Math. Bull. 54 (4) (2011), 716–725. | DOI | MR | Zbl

[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 | MR

[14] Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Progress in Math., vol. 118, Birkhäuser Verlag, Basel, 1994. | MR | Zbl

[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 | Zbl

Cité par Sources :