The method of averaging for Poisson connections on foliations and its applications. (English) Zbl 1456.53064

Summary: On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson fiber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.


53D17 Poisson manifolds; Poisson groupoids and algebroids
53C12 Foliations (differential geometric aspects)
53D20 Momentum maps; symplectic reduction
58D19 Group actions and symmetry properties
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
Full Text: DOI


[1] M. Avendaño-Camacho, J. A. Vallejo and Y. Vorobiev, Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems, J. Math. Phys., 54 (2013), 15pp. · Zbl 1302.37032
[2] M. Avendaño-Camacho and Y. Vorobiev, Deformations of Poisson structures on fibered manifolds and adiabatic slow-fast systems, Int. J. Geom. Methods Mod. Phys., 14 (2017), 15pp. · Zbl 1379.53097
[3] O. Brahic; R. L. Fernandes, Integrability and reduction of Hamiltonian actions on Dirac manifolds, Indag. Math., 25, 901-925 (2014) · Zbl 1298.53082
[4] O. Brahic and R. L. Fernandes, Poisson fibrations and fibered symplectic groupoids, in Poisson Geometry in Mathematics and Physics, Contemp. Math., 450, Amer. Math. Soc., Providence, RI, 2008, 41-59. · Zbl 1158.53064
[5] T. Courant and A. Weinstein, Beyond Poisson structures, in Action Hamiltoniennes de Groupes. Troisième Théorème de Lie, Travaux en Cours, 27, Hermann, Paris, 1988, 39-49. · Zbl 0698.58020
[6] T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319, 631-661 (1990) · Zbl 0850.70212
[7] J.-P. Dufour; A. Wade, On the local structure of Dirac manifolds, Compos. Math., 144, 774-786 (2008) · Zbl 1142.53063
[8] V. L. Ginzburg, Momentum mappings and Poisson cohomology, Internat. J. Math., 7, 329-358 (1996) · Zbl 0867.58028
[9] V. L. Ginzburg, Equivariant Poisson cohomology and a spectral sequence associated with a moment map, Internat. J. Math., 10, 977-1010 (1999) · Zbl 1061.53059
[10] I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. · Zbl 0782.53013
[11] J.-H. Lu, Momentum mappings and reduction of Poisson actions, in Symplectic Geometry, Groupoids, and Integrable Systems, Math. Sci. Res. Inst. Publ., 20, Springer, New York, NY, 1991,209-226. · Zbl 0735.58004
[12] J. Marsden, R. Montgomery and T. Ratiu, Reduction, symmetry, and phases in mechanics, Mem. Amer. Math. Soc., 88 (1990). · Zbl 0713.58052
[13] R. Montgomery, The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case, Commun. Math. Phys., 120, 269-294 (1988) · Zbl 0689.58043
[14] A. Pedroza; E. Velasco-Barreras; Y. Vorobiev, Unimodularity criteria for Poisson structures on foliated manifolds, Lett. Math. Phys., 108, 861-882 (2018) · Zbl 1387.53038
[15] M. R. Sepanski, Compact Lie Groups, Graduate Texts in Mathematics, 235, Springer, New York, 2007. · Zbl 1246.22001
[16] P. Ševera; A. Weinstein, Poisson geometry with a 3-form background. Noncommutative geometry and string theory, Progr. Theoret. Phys. Suppl., 144, 145-154 (2001) · Zbl 1029.53090
[17] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994. · Zbl 0810.53019
[18] I. Vaisman, Coupling Poisson and Jacobi structures on foliated manifolds, Int. J. Geom. Methods Mod. Phys., 1, 607-637 (2004) · Zbl 1079.53130
[19] I. Vaisman, Foliation-coupling Dirac structures, J. Geom. Phys., 56, 917-938 (2006) · Zbl 1109.53076
[20] J. A. Vallejo and Y. Vorobiev, Invariant Poisson realizations and the averaging of Dirac structures, SIGMA Symmetry Integrability Geom. Methods Appl., 10 (2014), 20pp. · Zbl 1301.53089
[21] Y. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, in Lie Algebroids and Related Topics in Differential Geometry, Banach Center Publ., 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001,249-274. · Zbl 1007.53062
[22] A. Wade, Poisson fiber bundles and coupling Dirac structures, Ann. Global Anal. Geom., 3, 207-217 (2008) · Zbl 1151.53070
[23] M. Wüstner, A connected Lie group equals the square of the exponential image, J. Lie Theory, 13, 307-309 (2003) · Zbl 1022.22005
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.