Foliation-coupling Dirac structures. (English) Zbl 1109.53076

The author shows that any Dirac structure is coupling with the fibers of a tubular neighborhood of an embedded presymplectic leaf.
The Dirac structures were first defined by Courant and Weinstein. In this paper, the author concentrates on the case where the manifold is endowed with a regular foliation and extends the notion of coupling from Poisson structures to Dirac structures.
Furthermore, the corresponding generalization of the Vorobjev characterization of coupling Poisson structures is obtained. The coupling condition is finally used along a submanifold, instead of a foliation. As a byproduct, an invariant which recalls the second fundamental form of a submanifold of a Riemannian manifold is found.


53D17 Poisson manifolds; Poisson groupoids and algebroids
Full Text: DOI arXiv


[1] Boucetta, M., Compatibilité des structures pseudo-riemanniennes et des structures de Poisson, CR acad. sci. Paris, Série I, 333, 763-768, (2001) · Zbl 1009.53057
[2] Bursztyn, H.; Crainic, M., Dirac structures, moment maps and quasi-Poisson manifolds, (), 1-40 · Zbl 1079.53123
[3] Courant, T.J.; Weinstein, A., Beyond Poisson structures, Actions hamiltoniennes de groupes. troisième théorème de Lie (Lyon 1986), (1988), Hermann Paris, pp. 39-49
[4] Courant, T.J., Dirac manifolds, Trans. am. math. soc., 319, 631-661, (1990) · Zbl 0850.70212
[5] Crainic, M.; Fernandez, R.L., Integrability of Poisson brackets, J. diff. geom., 66, 71-137, (2004) · Zbl 1066.53131
[6] Cruceanu, V.; Fortuny, P.; Gadea, P.M., A survey on paracomplex geometry, Rocky mountain J. math., 26, 1-33, (1996) · Zbl 0856.53049
[7] Dazord, P.; Lichnerowicz, A.; Marle, Ch.-M., Structures locales des variétés de Jacobi, J. math. pures appl., 70, 101-152, (1991) · Zbl 0659.53033
[8] Dorfman, I., Dirac structures and integrability of nonlinear evolution equation, (1993), John Wiley & Sons New York
[9] J.-P. Dufour, A. Wade, On the local structure of Dirac manifolds. arXiv:math.SG/0405257. · Zbl 1142.53063
[10] Hirsch, M.W., Differential topology, GTM 33, (1976), Springer-Verlag New York
[11] Itskov, V.M.; Karasev, M.; Vorobjev, Yu.M., Infinitesimal Poisson cohomology, Am. math. soc. transl., 187, 2, 327-360, (1998) · Zbl 0922.58028
[12] Liu, Z.-J.; Weinstein, A.; Xu, P., Manin triples for Lie bialgebroids, J. diff. geom., 45, 547-574, (1997) · Zbl 0885.58030
[13] Liu, Z.-J.; Weinstein, A.; Xu, P., Dirac structures and Poisson homogeneous spaces, Commun. math. phys., 192, 121-144, (1998) · Zbl 0921.58074
[14] MacKenzie, K.C.H., Lie algebroids and Lie pseudo-algebras, Bull. London math. soc., 27, 97-147, (1995) · Zbl 0829.22001
[15] Molino, P., Riemannian foliations, Progress in mathematics, vol. 73, (1988), Birkhäuser Boston
[16] D. Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds, Thesis, UC Berkeley, 1999.
[17] Vaisman, I., Dirac submanifolds of Jacobi manifolds, (), 603-622 · Zbl 1077.53066
[18] Vaisman, I., Coupling Poisson and Jacobi structures, Int. J. geom. meth. mod. phys., 1, 5, 607-637, (2004) · Zbl 1079.53130
[19] I. Vaisman, Transitive Courant algebroids. arXiv:math.DG/0407399. · Zbl 1159.53348
[20] Vorobjev, Yu.M., Coupling tensors and Poisson geometry near a single symplectic leaf, Lie algebroids and related topics in differential geometry, Banach center publ., Polish acad. sci. (Warsaw), 54, 249-274, (2001) · Zbl 1007.53062
[21] Wade, A., Conformal Dirac structures, Lett. math. phys., 53, 331-348, (2000) · Zbl 0982.53069
[22] Weinstein, A., The local structure of Poisson manifolds, J. diff. geom., 18, 523-557, (1983) · Zbl 0524.58011
[23] Xu, P., Dirac submanifolds and Poisson involutions, Ann. sci. ec. norm. sup., 36, 403-430, (2003) · Zbl 1047.53052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.