Splitting theorems for Poisson and related structures

Henrique Bursztyn; Hudson Lima; Eckhard Meinrenken

doi:10.1515/crelle-2017-0014

Article

Splitting theorems for Poisson and related structures

Henrique Bursztyn , Hudson Lima and Eckhard Meinrenken

Published/Copyright: March 18, 2017

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2019 Issue 754

Abstract

According to the Weinstein splitting theorem, any Poisson manifold is locally, near any given point, a product of a symplectic manifold with another Poisson manifold whose Poisson structure vanishes at the point. Similar splitting results are known, e.g., for Lie algebroids, Dirac structures and generalized complex structures. In this paper, we develop a novel approach towards these results that leads to various generalizations, including their equivariant versions as well as their formulations in new contexts.

Funding statement: Henrique Bursztyn and Hudson Lima thank Faperj and CNPq for financial support; Eckhard Meinrenken was supported by an NSERC Discovery Grant.

A Normal bundles of vector subbundles

For a vector bundle pr:E→M, with a vector subbundle F→N along a submanifold N⊆M, the normal bundle ν⁢(E,F) is a vector bundle over ν⁢(M,N) with projection ν⁢(pr):ν⁢(E,F)→ν⁢(M,N). In fact, ν⁢(E,F) fits into a double vector bundle

that is, in this diagram both horizontal and vertical arrows are vector-bundle projections, and the horizontal and vertical scalar multiplications commute (see [22] for this characterization of double vector bundles). In particular, for a submanifold N⊆M, we have a double vector bundle

The tangent bundle to ν⁢(M,N) also gives rise to a double vector bundle, the so-called tangent prolongation of p:ν⁢(M,N)→N:

Lemma A.1.

There is a natural map ν⁢(T⁢M,T⁢N)→T⁢ν⁢(M,N) which is a vector-bundle isomorphism with respect to the vector-bundle structures over TN and ν⁢(M,N), covering the identity map in each case. In particular, ν⁢(T⁢M,T⁢N) and T⁢ν⁢(M,N) are identified as double vector bundles.

Proof.

Let prM:T⁢M→M denote the tangent bundle to M. The iterated tangent bundle T⁢(T⁢M) is a double vector bundle

There is a canonical involution J:T⁢T⁢M→T⁢T⁢M satisfying (T⁢prM)∘J=prT⁢M and which interchanges the vertical and horizontal vector bundle structures. See, e.g., [30, Section 9.6].

For a submanifold N⊆M, the submanifolds T⁢(T⁢M|N) and (T⁢T⁢M)|T⁢N of TTM are both sub-double vector bundles:

The involution J:T⁢T⁢M→T⁢T⁢M restricts to an isomorphism T⁢(T⁢M|N)→(T⁢T⁢M)|T⁢N between these two double vector bundles. This map also restricts to the canonical involution of TTN, viewed as submanifolds of T⁢(T⁢M|N) and (T⁢T⁢M)|T⁢N. In this way, J gives rise to an isomorphism between the two double vector bundles

as desired. ∎

Acknowledgements

We would like to thank David Li-Bland, Pedro Frejlich, Ioan Mărcuţ and Shlomo Sternberg, for helpful comments on various aspects of our work. We also thank the anonymous referee for his/her comments and suggestions.

References

[1] M. Abouzaid and M. Boyarchenko, Local structure of generalized complex manifolds, J. Symplectic Geom. 4 (2006), no. 1, 43–62. 10.4310/JSG.2006.v4.n1.a2Search in Google Scholar

[2] R. Abraham, J. Marsden and T. Ratiu, Manifolds, tensor analysis and applications, Addison-Wesley, Reading 1983. Search in Google Scholar

[3] A. Alekseev and E. Meinrenken, Ginzburg–Weinstein via Gelfand–Zeitlin, J. Differential Geom. 76 (2007), no. 1, 1–34. 10.4310/jdg/1180135664Search in Google Scholar

[4] A. Alekseev and E. Meinrenken, Linearization of Poisson Lie group structures, J. Symplectic Geom. 14 (2016), no. 1, 227–267. 10.4310/JSG.2016.v14.n1.a9Search in Google Scholar

[5] I. Androulidakis and G. Skandalis, The holonomy groupoid of a singular foliation, J. reine angew. Math. 626 (2009), 1–37. 10.1515/CRELLE.2009.001Search in Google Scholar

[6] M. Bailey, G. Cavalcanti and J. van der Leer Duran, Blow-ups in generalized complex geometry, preprint (2016), https://arxiv.org/abs/1602.02076. 10.1090/tran/7412Search in Google Scholar

[7] R. Balan, A note about integrability of distributions with singularities, Boll. Unione Mat. Ital. A (7) 8 (1994), no. 3, 335–344. Search in Google Scholar

[8] J. Basto-Goncalves, Linearization of resonant vector fields, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6457–6476. 10.1090/S0002-9947-2010-04978-5Search in Google Scholar

[9] C. Blohmann, Removable presymplectic singularities and the local splitting of Dirac structures, preprint (2014), https://arxiv.org/abs/1410.5298; to appear in Int. Math. Res. Not. IMRN. 10.1093/imrn/rnw238Search in Google Scholar

[10] H. Bursztyn, A. Cabrera and M. del Hoyo, Vector bundles over Lie groupoids and algebroids, Adv. Math. 290 (2016), 163–207. 10.1016/j.aim.2015.11.044Search in Google Scholar

[11] I. Calvo and F. Falceto, Poisson reduction and branes in Poisson sigma models, Lett. Math. Phys. 70 (2004), 231–247. 10.1007/s11005-004-4302-7Search in Google Scholar

[12] A. S. Cattaneo and M. Zambon, Coisotropic embeddings in Poisson manifolds, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3721–3746. 10.1090/S0002-9947-09-04667-4Search in Google Scholar

[13] T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 2, 631–661. 10.1090/S0002-9947-1990-0998124-1Search in Google Scholar

[14] M. Crainic and I. Marcut, On the existence of symplectic realizations, J. Symplectic Geom. 9 (2011), no. 4, 435–444. 10.4310/JSG.2011.v9.n4.a2Search in Google Scholar

[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. 10.1007/s10455-011-9287-8Search in Google Scholar

[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. 10.4064/bc54-0-3Search in Google Scholar

[17] J.-P. Dufour and A. Wade, On the local structure of Dirac manifolds, Compos. Math. 144 (2008), no. 3, 774–786. 10.1112/S0010437X07003272Search in Google Scholar

[18] J.-P. Dufour and N. T. Zung, Poisson structures and their normal forms, Progr. Math. 242, Birkhäuser, Basel 2005. Search in Google Scholar

[19] R. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), no. 1, 119–179. 10.1006/aima.2001.2070Search in Google Scholar

[20] P. Frejlich and I. Marcut, The local normal form around Poisson transversals, preprint (2013), https://arxiv.org/abs/1306.6055; to appear in Pacific J. Math. Search in Google Scholar

[21] P. Frejlich and I. Marcut, Normal forms for Poisson maps and symplectic groupoids around Poisson transversals, preprint (2015), https://arxiv.org/abs/1508.05670. 10.1007/s11005-017-1007-2Search in Google Scholar PubMed PubMed Central

[22] J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59 (2009), no. 9, 1285–1305. 10.1016/j.geomphys.2009.06.009Search in Google Scholar

[23] M. Gualtieri, Generalized complex geometry, Ph.D. thesis, Oxford University, 2004. Search in Google Scholar

[24] M. Gualtieri, Generalized complex geometry, Ann. of Math. (2) 174 (2011), no. 1, 75–123. 10.4007/annals.2011.174.1.3Search in Google Scholar

[25] V. Guillemin and S. Sternberg, Geometric asymptotics, Math. Surveys Monogr. 14, American Mathematical Society, Providence 1990. Search in Google Scholar

[26] N. Hitchin, Generalized Calabi–Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281–308. 10.1093/qmath/hag025Search in Google Scholar

[27] S. Hu, Hamiltonian symmetries and reduction in generalized geometry, Houston J. Math. 35 (2009), no. 3, 787–811. Search in Google Scholar

[28] S. Lang, Differential and Riemannian Manifolds, Grad. Texts in Math. 160, Springer, Heidelberg 1995. 10.1007/978-1-4612-4182-9Search in Google Scholar

[29] D. Li-Bland and E. Meinrenken, Courant algebroids and Poisson geometry, Int. Math. Res. Not. IMRN 11 (2009), 2106–2145. 10.1093/imrn/rnp048Search in Google Scholar

[30] K. Mackenzie, General theory of Lie groupoids and Lie algebroids, London Math. Soc. Lecture Note Ser. 213, Cambridge University Press, Cambridge 2005. 10.1017/CBO9781107325883Search in Google Scholar

[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. 10.4310/MRL.2006.v13.n6.a14Search in Google Scholar

[32] D. Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds, Ph. D. thesis, University of California, Berkeley, 1999. Search in Google Scholar

[33] P. Ševera, Letters to Alan Weinstein, (1998–2000), available at http://sophia.dtp.fmph.uniba.sk/~severa/letters/. Search in Google Scholar

[34] P. Ševera, Poisson Lie T-duality and Courant algebroids, Lett. Math. Phys. 105 (2015), no. 12, 1689–1701. 10.1007/s11005-015-0796-4Search in Google Scholar

[35] P. Stefan, Integrability of systems of vector fields, J. Lond. Math. Soc. (2) 21 (1980), no. 3, 544–556. 10.1112/jlms/s2-21.3.544Search in Google Scholar

[36] S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 809–824. 10.2307/2372437Search in Google Scholar

[37] S. Sternberg, On the structure of local homeomorphisms of Euclidean n-space. II, Amer. J. Math. 80 (1958), 623–631. 10.2307/2372774Search in Google Scholar

[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. 10.1073/pnas.74.12.5253Search in Google Scholar PubMed PubMed Central

[39] H. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–188. 10.1090/S0002-9947-1973-0321133-2Search in Google Scholar

[40] A. Weinstein, A universal phase space for particles in Yang–Mills fields, Lett. Math. Phys. 2 (1978), 417–420. 10.1007/BF00400169Search in Google Scholar

[41] A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), no. 3, 523–557. 10.4310/jdg/1214437787Search in Google Scholar

[42] A. Weinstein, Almost invariant submanifolds for compact group actions, J. Eur. Math. Soc. (JEMS) 2 (2000), no. 1, 53–86. 10.1007/s100970050014Search in Google Scholar

[43] P. Xu, Dirac submanifolds and Poisson involutions, Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), no. 3, 403–430. 10.1016/S0012-9593(03)00013-2Search in Google Scholar

Received: 2016-07-19

Revised: 2017-01-12

Published Online: 2017-03-18

Published in Print: 2019-09-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/crelle-2017-0014