Skip to main content
SpringerLink
Log in

Rigidity around Poisson submanifolds

  • Published:
Acta Mathematica

Abstract

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash–Moser fast convergence method. In the case of one-point submanifolds (fixed points), this implies a stronger version of Conn’s linearization theorem [2], also proving that Conn’s theorem is a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem [7]. We can also use the rigidity theorem to compute the Poisson moduli space of the sphere in the dual of a compact semisimple Lie algebra [17].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

  2. Conn J.F.: Normal forms for smooth Poisson structures. Ann. of Math., 121, 565–593 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Crainic, M. & Fernandes, R. L., Integrability of Lie brackets. Ann. of Math., 157 (2003), 575–620.

  4. Crainic, M. & Fernandes, R. L., Rigidity and flexibility in Poisson geometry, in Travaux mathématiques. Fasc. XVI, pp. 53–68. University of Luxembourg, Luxembourg, 2005.

  5. Crainic, M. & Fernandes, R. L., Stability of symplectic leaves. Invent. Math., 180 (2010), 481–533.

  6. Crainic, M. & Fernandes, R. L., A geometric approach to Conn’s linearization theorem. Ann. of Math., 173 (2011), 1121–1139.

  7. Crainic, M. & Mărcuţ, I., A normal form theorem around symplectic leaves. J. Differential Geom., 92 (2012), 417–461.

  8. Duistermaat, J. J. & Kolk, J. A. C., Lie Groups. Universitext. Springer, Berlin– Heidelberg, 2000.

  9. Fernandes, R. L., Ortega, J. P. & Ratiu, T. S., The momentum map in Poisson geometry. Amer. J. Math., 131 (2009), 1261–1310.

  10. Gilkey, P. B., Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem. Mathematics Lecture Series, 11. Publish or Perish, Wilmington, DE, 1984.

  11. Guillemin, V. & Sternberg, S., A normal form for the moment map, in Differential Geometric Methods in Mathematical Physics (Jerusalem, 1982), Math. Phys. Stud., 6, pp. 161–175. Reidel, Dordrecht, 1984.

  12. Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc., 7, 65–222 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mackenzie, K. C. H., General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series, 213. Cambridge University Press, Cambridge, 2005.

  14. Mackenzie, K. C. H. & Xu, P., Integration of Lie bialgebroids. Topology, 39 (2000), 445– 467.

  15. Mărcuţ, I., Formal equivalence of Poisson structures around Poisson submanifolds. Pacific J. Math., 255 (2012), 439–461.

  16. Mărcuţ, I., Normal Forms in Poisson Geometry. Ph.D. Thesis, Utrecht University, Utrecht, 2013. arXiv:1301.4571 [math.DG].

  17. Mărcuţ, I., Deformations of the Lie–Poisson sphere of a compact semisimple Lie algebra. Compos. Math., 150 (2014), 568–578.

  18. Miranda, E., Monnier, P. & Zung, N. T., Rigidity of Hamiltonian actions on Poisson manifolds. Adv. Math., 229 (2012), 1136–1179.

  19. Moerdijk, I. & Mrčun, J., Introduction to Foliations and Lie Groupoids. Cambridge Studies in Advanced Mathematics, 91. Cambridge University Press, Cambridge, 2003.

  20. Monnier, P. & Zung, N. T., Levi decomposition for smooth Poisson structures. J. Differential Geom., 68 (2004), 347–395.

  21. Montgomery R.: Canonical formulations of a classical particle in a Yang–Mills field and Wong’s equations. Lett. Math. Phys., 8, 59–67 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pflaum, M. J., Posthuma, H. & Tang, X., Geometry of orbit spaces of proper Lie groupoids. To appear in J. Reine. Angew. Math. arXiv:1101.0180 [math.DG].

  23. Vorobjev, Y., Coupling tensors and Poisson geometry near a single symplectic leaf, in Lie Algebroids and Related Topics in Differential Geometry (Warsaw, 2000), Banach Center Publications, 54, pp. 249–274. Polish Acad. Sci. Inst. Math., Warsaw, 2001.

  24. Weinstein A.: The local structure of Poisson manifolds. J. Differential Geom., 18, 523–557 (1983)

    MathSciNet  MATH  Google Scholar 

  25. Weinstein A.: Poisson geometry of the principal series and nonlinearizable structures. J. Differential Geom., 25, 55–73 (1987)

    MathSciNet  MATH  Google Scholar 

  26. Weinstein, A. & Xu, P., Extensions of symplectic groupoids and quantization. J. Reine Angew. Math., 417 (1991), 159–189.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801, U.S.A

    Ioan Mărcuţ

Authors
  1. Ioan Mărcuţ
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ioan Mărcuţ.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mărcuţ, I. Rigidity around Poisson submanifolds. Acta Math 213, 137–198 (2014). https://doi.org/10.1007/s11511-014-0118-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11511-014-0118-1

Keywords

Search

Navigation