×

zbMATH — the first resource for mathematics

Splitting theorems for Poisson and related structures. (English) Zbl 1425.53103
This paper gives a new approach of the Weinstein splitting theorem, that leads to various generalizations. First, the authors discuss the linearization of Euler-like vector fields and the resulting tubular neighborhood embeddings. Then, they apply this to anchored vector bundles and obtain a normal form theorem along transversals. They also obtain similar transversal normal form theorems in new contexts. This work includes also equivariant versions of the results.

MSC:
53D17 Poisson manifolds; Poisson groupoids and algebroids
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] M. Abouzaid and M. Boyarchenko, Local structure of generalized complex manifolds, J. Symplectic Geom. 4 (2006), no. 1, 43-62. · Zbl 1116.53055
[2] R. Abraham, J. Marsden and T. Ratiu, Manifolds, tensor analysis and applications, Addison-Wesley, Reading 1983. · Zbl 0875.58002
[3] A. Alekseev and E. Meinrenken, Ginzburg-Weinstein via Gelfand-Zeitlin, J. Differential Geom. 76 (2007), no. 1, 1-34. · Zbl 1119.53053
[4] A. Alekseev and E. Meinrenken, Linearization of Poisson Lie group structures, J. Symplectic Geom. 14 (2016), no. 1, 227-267. · Zbl 1357.53096
[5] I. Androulidakis and G. Skandalis, The holonomy groupoid of a singular foliation, J. reine angew. Math. 626 (2009), 1-37. · Zbl 1161.53020
[6] M. Bailey, G. Cavalcanti and J. van der Leer Duran, Blow-ups in generalized complex geometry, preprint (2016), . · Zbl 1409.53067
[7] R. Balan, A note about integrability of distributions with singularities, Boll. Unione Mat. Ital. A (7) 8 (1994), no. 3, 335-344. · Zbl 0852.57024
[8] J. Basto-Goncalves, Linearization of resonant vector fields, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6457-6476. · Zbl 1221.32015
[9] C. Blohmann, Removable presymplectic singularities and the local splitting of Dirac structures, preprint (2014), ; to appear in Int. Math. Res. Not. IMRN. · Zbl 1405.53112
[10] H. Bursztyn, A. Cabrera and M. del Hoyo, Vector bundles over Lie groupoids and algebroids, Adv. Math. 290 (2016), 163-207. · Zbl 1391.53092
[11] I. Calvo and F. Falceto, Poisson reduction and branes in Poisson sigma models, Lett. Math. Phys. 70 (2004), 231-247. · Zbl 1078.81067
[12] A. S. Cattaneo and M. Zambon, Coisotropic embeddings in Poisson manifolds, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3721-3746. · Zbl 1169.53060
[13] T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 2, 631-661. · Zbl 0850.70212
[14] M. Crainic and I. Marcut, On the existence of symplectic realizations, J. Symplectic Geom. 9 (2011), no. 4, 435-444. · Zbl 1244.53091
[15] L. Drager, J. Lee, E. Park and K. Richardson, Smooth distributions are finitely generated, Ann. Global Anal. Geom. 41 (2012), no. 3, 357-369. · Zbl 1243.57018
[16] J.-P. Dufour, Normal forms for Lie algebroids, Lie algebroids and related topics in differential geometry (Warsaw 2000), Banach Center Publ. 54, Polish Academy of Sciences, Warsaw (2001), 35-41. · Zbl 1004.53019
[17] J.-P. Dufour and A. Wade, On the local structure of Dirac manifolds, Compos. Math. 144 (2008), no. 3, 774-786. · Zbl 1142.53063
[18] J.-P. Dufour and N. T. Zung, Poisson structures and their normal forms, Progr. Math. 242, Birkhäuser, Basel 2005.
[19] R. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), no. 1, 119-179. · Zbl 1007.22007
[20] P. Frejlich and I. Marcut, The local normal form around Poisson transversals, preprint (2013), ; to appear in Pacific J. Math.
[21] P. Frejlich and I. Marcut, Normal forms for Poisson maps and symplectic groupoids around Poisson transversals, preprint (2015), . · Zbl 1387.53106
[22] J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59 (2009), no. 9, 1285-1305. · Zbl 1171.58300
[23] M. Gualtieri, Generalized complex geometry, Ph.D. thesis, Oxford University, 2004.
[24] M. Gualtieri, Generalized complex geometry, Ann. of Math. (2) 174 (2011), no. 1, 75-123. · Zbl 1235.32020
[25] V. Guillemin and S. Sternberg, Geometric asymptotics, Math. Surveys Monogr. 14, American Mathematical Society, Providence 1990.
[26] N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281-308. · Zbl 1076.32019
[27] S. Hu, Hamiltonian symmetries and reduction in generalized geometry, Houston J. Math. 35 (2009), no. 3, 787-811. · Zbl 1183.53077
[28] S. Lang, Differential and Riemannian Manifolds, Grad. Texts in Math. 160, Springer, Heidelberg 1995.
[29] D. Li-Bland and E. Meinrenken, Courant algebroids and Poisson geometry, Int. Math. Res. Not. IMRN 11 (2009), 2106-2145. · Zbl 1169.53061
[30] K. Mackenzie, General theory of Lie groupoids and Lie algebroids, London Math. Soc. Lecture Note Ser. 213, Cambridge University Press, Cambridge 2005. · Zbl 1078.58011
[31] E. Miranda and N. T. Zung, A note on equivariant normal forms of Poisson structures, Math. Res. Lett. 13 (2006), no. 5-6, 1001-1012. · Zbl 1112.53062
[32] D. Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds, Ph. D. thesis, University of California, Berkeley, 1999.
[33] P. Ševera, Letters to Alan Weinstein, (1998-2000), available at .
[34] P. Ševera, Poisson Lie T-duality and Courant algebroids, Lett. Math. Phys. 105 (2015), no. 12, 1689-1701. · Zbl 1344.53064
[35] P. Stefan, Integrability of systems of vector fields, J. Lond. Math. Soc. (2) 21 (1980), no. 3, 544-556. · Zbl 0432.58002
[36] S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 809-824. · Zbl 0080.29902
[37] S. Sternberg, On the structure of local homeomorphisms of Euclidean n-space. II, Amer. J. Math. 80 (1958), 623-631. · Zbl 0083.31406
[38] S. Sternberg, On minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field, Proc. Natl. Acad. Sci. USA 74 (1977), 5253-5254. · Zbl 0765.58010
[39] H. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171-188. · Zbl 0274.58002
[40] A. Weinstein, A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys. 2 (1978), 417-420. · Zbl 0388.58010
[41] A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), no. 3, 523-557. · Zbl 0524.58011
[42] A. Weinstein, Almost invariant submanifolds for compact group actions, J. Eur. Math. Soc. (JEMS) 2 (2000), no. 1, 53-86. · Zbl 0957.53021
[43] P. Xu, Dirac submanifolds and Poisson involutions, Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), no. 3, 403-430. · 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.