Geodesic
$G$-Invariant Deformations of Almost-Coupling Poisson Structures
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

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On a foliated manifold equipped with an action of a compact Lie group $G$, we study a class of almost-coupling Poisson and Dirac structures, in the context of deformation theory and the method of averaging.
Keywords: Poisson geometry; Dirac structures; deformation; averaging. 
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     author = {Jos\'e Antonio Vallejo and Yury Vorobiev},
     title = {$G${-Invariant} {Deformations} of {Almost-Coupling} {Poisson} {Structures}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2017},
     volume = {13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a21/}
}
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José Antonio Vallejo; Yury Vorobiev. $G$-Invariant Deformations of Almost-Coupling Poisson Structures. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a21/

[1] Arnold V. I., Kozlov V. V., Neishtadt A. I., Dynamical systems, v. III, Encyclopaedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 1988 | DOI | MR | Zbl

[2] Avendaño Camacho M., Vallejo J. A., Vorobiev Yu., “Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems”, J. Math. Phys., 54 (2013), 082704, 15 pp., arXiv: 1305.3974 | DOI | MR | Zbl

[3] Avendaño Camacho M., Vallejo J. A., Vorobiev Yu., “A simple global representation for second-order normal forms of Hamiltonian systems relative to periodic flows”, J. Phys. A: Math. Theor., 46 (2013), 395201, 12 pp. | DOI | MR | Zbl

[4] Avendaño Camacho M., Vorobiev Yu., “On deformations of Poisson structures on fibered manifolds and adiabatic slow-fast systems”, Int. J. Geom. Methods Mod. Phys., 14 (2017), 1750086, 15 pp. | DOI

[5] Brahic O., Fernandes R. L., “Poisson fibrations and fibered symplectic groupoids”, Poisson Geometry in Mathematics and Physics, Contemp. Math., 450, Amer. Math. Soc., Providence, RI, 2008, 41–59, arXiv: math.DG/0702258 | DOI | MR | Zbl

[6] Bursztyn H., Radko O., “Gauge equivalence of Dirac structures and symplectic groupoids”, Ann. Inst. Fourier (Grenoble), 53 (2003), 309–337, arXiv: math.SG/0202099 | DOI | MR | Zbl

[7] Courant T. J., “Dirac manifolds”, Trans. Amer. Math. Soc., 319 (1990), 631–661 | DOI | MR | Zbl

[8] Courant T. J., Weinstein A., “Beyond Poisson structures”, Action hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986), Travaux en Cours, 27, Hermann, Paris, 1988, 39–49 | MR

[9] Crainic M., Fernandes R. L., “Stability of symplectic leaves”, Invent. Math., 180 (2010), 481–533, arXiv: 0810.4437 | DOI | MR | Zbl

[10] Crainic M., Mărcuţ I., “A normal form theorem around symplectic leaves”, J. Differential Geom., 92 (2012), 417–461, arXiv: 1009.2090 | DOI | MR | Zbl

[11] Frejlich P., Mărcuţ I., The normal form theorem around Poisson transversals, arXiv: 1306.6055

[12] Guillemin V., Lerman E., Sternberg S., Symplectic fibrations and multiplicity diagrams, Cambridge University Press, Cambridge, 1996 | DOI | MR | Zbl

[13] Kolář I., Michor P. W., Slovák J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993 | DOI | MR | Zbl

[14] Ševera P., Weinstein A., “Poisson geometry with a 3-form background”, Progr. Theoret. Phys. Suppl., 2001, 145–154, arXiv: math.SG/0107133 | DOI | MR

[15] Vaisman I., Lectures on the geometry of Poisson manifolds, Progress in Mathematics, 118, Birkhäuser Verlag, Basel, 1994 | DOI | MR | Zbl

[16] Vaisman I., “Coupling Poisson and Jacobi structures on foliated manifolds”, Int. J. Geom. Methods Mod. Phys., 1 (2004), 607–637, arXiv: math.SG/0402361 | DOI | MR | Zbl

[17] Vaisman I., “Foliation-coupling Dirac structures”, J. Geom. Phys., 56 (2006), 917–938, arXiv: math.SG/0412318 | DOI | MR | Zbl

[18] Vallejo J. A., Vorobiev Yu., “Invariant Poisson realizations and the averaging of Dirac structures”, SIGMA, 10 (2014), 096, 20 pp., arXiv: 1405.0574 | DOI | MR | Zbl

[19] Vorobiev Yu., “Averaging of Poisson structures”, Geometric Methods in Physics, AIP Conf. Proc., 1079, Amer. Inst. Phys., Melville, NY, 2008, 235–240 | DOI | MR | Zbl

[20] Vorobjev Yu., “Coupling tensors and Poisson geometry near a single symplectic leaf”, Lie Algebroids and Related Topics in Differential Geometry (Warsaw, 2000), Banach Center Publ., 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001, 249–274, arXiv: math.SG/0008162 | DOI | MR | Zbl

[21] Wade A., “Poisson fiber bundles and coupling Dirac structures”, Ann. Global Anal. Geom., 33 (2008), 207–217, arXiv: math.SG/0507594 | DOI | MR | Zbl