Abstract
In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a b-Poisson manifold as we will see in [9].
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References
M. Bertelson. Foliations associated to regular Poisson structures. Commun. Contemp. Math., 3(3) (2001), 441–456.
C. Camacho and A. Lins Neto. Geometric theory of foliations. Birkhauser (1985).
M. Crainic and R.L. Fernandes. Integrability of Poisson brackets. J. of Differential Geometry, 66 (2004), 71–137.
M. Crainic and R.L. Fernandes. Integrability of Lie brackets. Annals of Mathematics, 157 (2003), 575–620.
S. Evens, J-H. Lu and A. Weinstein. Transverse measures, the modular class and a cohomology pairing for Lie algebroids. Quart. J. Math. Oxford Ser. (2), 50(200) (1999), 417–436.
M. Gotay. On coisotropic embeddings of presymplectic manifolds. Proc. Amer. Math. Soc., 84 (1982), 111–114.
M. Gotay, R. Lashof, J. Sniaticky and A. Weinstein. Closed forms on symplectic fibre bundles. Comment. Math. Helvetici, 58 (1983), 617–621.
V. Guillemin, E. Lerman and S. Sternberg. Symplectic fibrations and multiplicity diagrams. Cambridge University Press (1996).
V. Guillemin, E. Miranda and A.R. Pires. Symplectic and Poisson geometry of b-manifolds, preprint (2010).
Y. Kosmann-Schwarzbach, C. Laurent-Gengoux and A. Weinstein. Modular classes of Lie algebroid morphisms. Transform. Groups, 13(3–4) (2008), 727–755.
Y. Kosmann-Schwarzbach. Poisson manifolds, Lie algebroids, modular classes: a survey. SIGMA, 4 (2008), 005, 30 pp.
H. Li. Topology of co-symplectic/co-kahler manifolds. Asian. J. Math., 12(4) (2008), 527–544.
P. Libermann. Sur les automorphismes infinitésimaux des structures symplectiques et des structures de contact. 1959 Colloque Géom. Diff. Globale (Bruxelles, 1958) pp. 37–359 Centre Belge Rech. Math., Louvain.
V. Mamaev. Codimension one foliations of flat 3-manifolds. Sbornik: Mathematics, 187(6) (1996), 823–833.
R. Melrose. Atiyah-Patodi-Singer Index Theorem. A.K. Peters, Wellesley (1993).
G. Reeb. Sur certaines propriétés toplogiques des variétés. Actualités Sci. Indust., 1183 (1952), Paris: Hermann.
W.P. Thurston. Existence of codimension-one foliations. Ann. of Math. (2), 104(2) (1976), 249–268.
D. Tischler. On fibering certain foliated manifolds over S1. Topology, 9 (1970), 153–154.
A. Weinstein. The modular automorphism group of a Poisson manifold. J. Geom. and Phys., 23(3–4) (1997), 379–394.
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Dedicated to the memory of Paulette Libermann whose cosymplectic manifolds play a fundamental role in this paper
Eva Miranda is partially supported by the DGICYT/FEDER project MTM2009-07594: Estructuras Geometricas: Deformaciones, Singularidades y Geometria Integral.
Ana Rita Pires was partially supported by a grant SFRH/BD/21657/2005 of Fundação para a Ciência e Tecnologia.
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Guillemin, V., Miranda, E. & Pires, A.R. Codimension one symplectic foliations and regular Poisson structures. Bull Braz Math Soc, New Series 42, 607–623 (2011). https://doi.org/10.1007/s00574-011-0031-6
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DOI: https://doi.org/10.1007/s00574-011-0031-6