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Compatible poisson structures on fibered 5-manifolds

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We study a class of Poisson tensors on a fibered manifold which are compatible with the fiber bundle structure by the so-called almost coupling condition. In the case of a 5-dimensional orientable fibered manifolds with 2-dimensional bases, we describe a global behavior of almost coupling Poisson tensors and their singularities by using a bigraded factorization of the Jacobi identity. In particular, we present some unimodularity criteria and describe a class of gauge type transformations preserving the almost coupling property.

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  1. 1.

    Avendaño-Camacho, M., Vorobiev, Yu.: Deformations of Poisson structures on fibered manifolds and adiabatic slow-fast systems. Int. J. Geom. Methods Mod. Phys. 14, 1750086 (2017)

  2. 2.

    Brahic, O., Fernandes, R.L.: Poisson fibrations and fibered symplectic groupoids. In: Poisson Geometry in Mathematics and Physics, Contemp. Math. 450, 41–59, (Amer. Math. Soc., Providence, RI, 2008)

  3. 3.

    Bursztyn, H., Radko, O.: Gauge equivalence of Dirac structures and symplectic groupoids. Ann. Inst. Fourier (Grenoble) 53, 309–337 (2003)

  4. 4.

    Carinena, J.F., Ibort, A., Marmo, G., Perelomov, A.M.: On the geometry of Lie algebras and Poisson tensors. J. Phys. A Math. Gen. 27, 7425–49 (1994)

  5. 5.

    Crainic, M., Marcut, I.: A normal form theorem around symplectic leaves. J. Differ. Geom. 92, 417–461 (2012)

  6. 6.

    Damianou, P.A., Petalidou, F.: Poisson brackets with prescribed Casimirs. Can. J. Math. 64, 991–1018 (2012)

  7. 7.

    Evangelista-Alvarado, M., Suárez-Serrato, P., Torres-Orozco, J., Vera, R.: On Bott-Morse Foliations and their Poisson Structures in Dimension 3, arXiv:1801.09735 [math.SG]

  8. 8.

    Flores-Espinoza, R.: On Poisson structures on \({\mathbb{R}}^{4}\), arXiv:1306.5254 [math-ph]

  9. 9.

    Garcia-Naranjo, L.C., Suárez-Serrato, P., Vera, R.: Poisson structures on smooth 4-manifolds. Lett. Math. Phys. 105, 1533–1550 (2015)

  10. 10.

    Grabowski, J., Marmo, G., Perelomov, A.M.: Poisson structures: towards a classification. Mod. Phys. Lett. A 18, 1719–33 (1993)

  11. 11.

    Gumral, H., Nutku, Y.: Poisson structures of dynamical systems with three degrees of freedom. J. Math. Phys. 34, 5691–5723 (1993)

  12. 12.

    Liu, Z.-J., Xu, P.: On quadratic Poisson structures. Lett. Math. Phys. 26, 33–42 (1992)

  13. 13.

    Marcut, I.: Rigidity around Poisson submanifolds. Acta. Math. 213, 137–198 (2014)

  14. 14.

    Marsden, J.E., Montgomery, R., Ratiu, T.: Reduction, symmetry and phases in mechanics. In: Mem. Am. Math. Soc. (436) 88, 1–110, (Amer. Math. Soc., Providence, RI, 1990)

  15. 15.

    Montgomery, R., Marsden, J.E., Ratiu, T.: Gauged Lie-Poisson structures. In: Fluids and Plasmas: Geometry and Dynamics, Cont. Math., eds. J. E. Marsden 28, 101–114, (Amer. Math. Soc., Boulder, CO, 1984)

  16. 16.

    Pedroza, A., Velasco-Barreras, E., Vorobiev, Yu.: Unimodularity criteria for Poisson structures on foliated manifolds. Lett. Math. Phys. 108, 861–882 (2018)

  17. 17.

    Radko, O.: A classification of topologically stable Poisson structures on a compact oriented surface. J. Symplectic Geom. 1, 523–542 (2002)

  18. 18.

    Severa, P., Weinstein, A.: Poisson geometry with a 3-form background. Progr. Theor. Phys. Suppl. 144, 145–154 (2001)

  19. 19.

    Suárez-Serrato, P., Torres-Orozco, J.: Poisson structures on wrinkled fibrations. Bol. Soc. Mat. Mex. 22, 263–280 (2016)

  20. 20.

    Vaisman, I.: Lectures on the Geometry of Poisson Manifolds, vol. 206. Birkhüuser Basel, Boston (1994)

  21. 21.

    Vaisman, I.: Coupling Poisson and Jacobi structures on foliated manifolds. Int. J. Geom. Methods Mod. Phys. 1, 607–637 (2004)

  22. 22.

    Vallejo, J.A., Vorobiev, Yu.: \(G\)-Invariant deformations of almost coupling Poisson structures. SIGMA 13, 22 (2017)

  23. 23.

    Vorobjev, Yu.: Coupling tensors and Poisson geometry near a single symplectic leaf, In Lie Algebroids and Related Topics in Differential Geometry (Warsaw, 2000), Banach Center Publ. 54, 249-274, Polish Acad. Sci. Inst. Math., Waszawa (2001)

  24. 24.

    Wade, A.: Poisson fiber bundles and coupling Dirac structures. Ann. Glob. Anal. Geom. 33, 207–217 (2008)

  25. 25.

    Weinstein, A.: The modular automorphism group of a Poisson manifold. J. Geom. Phys. 23, 379–394 (1997)

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This work was partially supported by the Mexican National Council of Science and Technology (CONACyT), under research project CB-2013-219631.

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Correspondence to J. C. Ruíz-Pantaleón.

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Flores-Espinoza, R., Ruíz-Pantaleón, J.C. & Vorobiev, Y. Compatible poisson structures on fibered 5-manifolds. Bol. Soc. Mat. Mex. 26, 187–209 (2020).

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  • Poisson structures
  • Fiber bundles
  • Almost coupling tensors
  • Poisson connections

Mathematics Subject Classification

  • 53D17
  • 53C12
  • 70G45