Abstract
We study various properties of polarized Poisson structure subordinate to the polarized symplectic structure.We study also the notion of a polarized Poisson manifold, i.e., a Poisson manifold foliated by coisotropic submanifolds. We show that the characteristic distribution of a polarized Poisson structure is completely integrable and its leaves are symplectic. In the particular case when the foliation is Lagrangian, we show that, the polarized Poisson manifold is also foliated by polarized symplectic leaves, and we prove Darboux’s theorem corresponding to a Lagrangian foliation with respect to a polarized Poisson manifold.
Similar content being viewed by others
References
Awane, A.: k-symplectic structures. J. Math. Phys. 33, 4046–4052 (1992)
Awane, A.: Generalized polarized manifolds. Rev. Mat. Complut. 21(1), 251–264 (2007)
Awane, A., Goze, M.: K-symplectic systems. Pfaffian systems. Kluwer Academic Press, New York (2000)
da Silva, A. C., Weinstein, A: Geometric models for noncommutative algebras. American Mathematical Society Berkeley Center for Pure and Applied Mathematics, vol. 10, Providence (1999)
Dazord, P.: Sur la géométrie des sons-fibrés et des feuilletages Lagrangiens. Ann. Ecole Normale Sup. 14, 465–480 (1981)
Fernandes, R.L., Laurent-Gengoux, C., Vanhaecke, P.: Global action-angle variables for non-commutative integrable systems. J. Symplectic Geom. 16(3), 645–699 (2018)
Guillemin, V., Sternberg, S.: Symplectic techniques in physics. Cambridge University Press, Cambridge (1984)
Laurent - Gengoux, C., Miranda, E., Vanhaecke, P.: Action-angle coordinates for integrable systems on Poisson manifolds. Int. Math. Res. Not. 2011(8), 1839–1869 (2011)
Libermann, P., Marle, C.M.: Symplectic geometry and analytical mechanics. U.E.R. de Mathématiques, L.A. 212 et E.R.A. 944, 1020, 1021 du C.N.R.S (2012)
Libermann, P.: Problèmes déquivalence et géométrie symplectique Astérisque, tome, pp. 107–108, 43–68 (1983)
Lichnerowicz, A.: Les variétés de Poisson et feuilletages. Ann. de la Faculté des Sci. de Toulouse Math. Série 5 Tome 4(3–4), 195–262 (1982)
Lichnerowicz, A.: Les variétés de Poisson et leurs algèbres de Lie associées. J. Differ. Geom. 13, 253–300 (1977)
Mitric, G., Vaisman, I.: Poisson structures on tangent bundles. Differ. Geom. Appl. 18, 207–228 (2003)
Molino, P.: Géométrie des Polarisations. In: Feuilletages et quantification géométrique, Travaux en cours, Hermann, Paris, pp. 37–53 (1984)
Molino, P.: Riemannian foliations, progress in mathematics, vol. 73. Birkhauser, Boston (1988) [MR932463 (89b:53054)]
Tamura, I., Sato, A.: On transverse foliations. Publi. Math. de I.H.É. S. Tome 54, 5–35 (1981)
Vaisman, I.: Lectures on the geometry of Poisson manifolds. Birkhauser, Berlin, Basel, Boston (1994)
Weinstein, A.: Lectures on symplectic manifolds. In: Conference board of the mathematical sciences No. 29, American Mathematical Society, Providence, RI (1976)
Weinstein, A.: The local structure of Poisson manifolds. J. Differ. Geom. 18, 523–557 (1983)
Weinstein, A.: Coisotropic calculus and Poisson groupoids. J. Math. Soc. Jpn. 40(4), 705–727 (1988)
Woodhouse, N.M.J.: Geometric quantization. Clarendon, Oxford (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Awane, A., Ismail, B. & Souhaila, E.A. Polarized Poisson structures. Beitr Algebra Geom 62, 799–813 (2021). https://doi.org/10.1007/s13366-020-00540-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13366-020-00540-5