Advertisement

The Geometry of Partial Order on Contact Transformations of Prequantization Manifolds

  • Gabi Ben Simon
Part of the Progress in Mathematics book series (PM, volume 279)

Summary

In this chapter we find a connection between the Hofer metric of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold, with an integral symplectic form, and the geometry, defined in (12), of the quantomorphisms group of its prequantization manifold. This gives two main results: First, we calculate, partly, the geometry of the quantomorphism group of a prequantization manifold of an integral symplectic manifold which admits a certain Lagrangian foliation. Second, for every prequantization manifold we give a formula for the distance between a point and a distinguished curve in the metric space associated to its group of quantomorphisms. Moreover, our first result is a full computation of the geometry related to the symplectic linear group which can be considered as a subgroup of the contactomorphism group of suitable prequantization manifolds of complex projective space. In the course of the proof we use in an essential way the Maslov quasimorphism.

Keywords

Partial Order Universal Cover Symplectic Manifold Contact Structure Hamiltonian Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Abraham,R.,Marsden,J. Foundations of Mechanics, second edition, 1994. Addison-Wesley Publishing Company.Google Scholar
  2. [2]
    Arnold,V.I. Mathematical Methods of Classical Mechanics, second edition, Graduate Texts in Mathematics; 60, 1989.Google Scholar
  3. [3]
    Arnold,V.I.,Givental,A. Symplectic Geometry, in, Encyclopaedia of Mathematical Sciences, Vol. 4, Dynamical systems IV, V. Arnold, S. Novikov eds., Springer, 1990, pp. 1-136.Google Scholar
  4. [4]
    Banyaga, A.,Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comm. Math. Helv. 53:2 (1978), 174-227.Google Scholar
  5. [5]
    Barge,J.,Ghys,E. Cocycle D’Euler et de Maslov Math.Ann. 294:2 (1992)235–265.CrossRefMathSciNetGoogle Scholar
  6. [6]
    Bates,S.,Weinstein,A. Lectures on the Geometry of Quantization. AMS, Berkeley Center for Pure and Applied Mathematics, 1997.MATHGoogle Scholar
  7. [7]
    Bavard, C. Longueur stable des commutateurs, L’Enseign. Math. 37:1-2 (1991), 109–150.MATHMathSciNetGoogle Scholar
  8. [8]
    Bott,R.,Tu,L.W. Differential Forms in algebraic Topology, Springer, 1982.Google Scholar
  9. [9]
    Carter,R.,Segal,G.,MacDonald,I. Lectures on Lie Groups and Lie Algebras. London Mathematical Society Student Texts 32, Cambridge University Press.Google Scholar
  10. [10]
    Dupont,J. Bounds for the chercteristic numbers of flat bundles. Notes Math.,vol 763 p.109-119 Berlin, Heidelberg, New York, Springer, 1979.Google Scholar
  11. [11]
    Eliashberg,Y., Kim,S.K., Polterovich,P. Geometry of contact transformations and domains: orderability vs. squeezing, Preprint math.SG/0511658, 2005.Google Scholar
  12. [12]
    Eliashberg,Y.;Polterovich,L. Partially ordered groups and geometry of contact transformations. Geom.Funct.Anal., 10(2000), 6, 1448-1476.Google Scholar
  13. [13]
    Givental A.B. Nonlinear generalization of the Maslov index. Theory of singularities and its applications, Adv. Soviet Math., vol,1. American Mathematical Society, Rhode Island,1990, p.71-103.Google Scholar
  14. [14]
    Hall,B.C. Lie groups, Lie Algebras, and Representations: An Elementary Introduction., Graduate Texts in Mathematics; 222, Springer, 2003.Google Scholar
  15. [15]
    Hofer,H.,Zehnder,E. Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Advanced Texts, Birkhäuser Verlag, 1994.MATHGoogle Scholar
  16. [16]
    Kirillov,A. Geometric quantization, in Encyclopaedia of Mathematical Sciences, Vol. 4, Dynamical systems IV, V. Arnold, S. Novikov eds., Springer, 1990, pp. 137-172.Google Scholar
  17. [17]
    McDuff,D., Salamon, D., Introduction to symplectic topology, second edition, Oxford University Press, Oxford, 1998.MATHGoogle Scholar
  18. [18]
    Nakahara,M. Geometry, Topology and Physics, Institute of Physics Publishing Bristol and Philadelphia, 1990.Google Scholar
  19. [19]
    Olshanskii,G.I. Invariant orderings in simple Lie groups. The solution to E.B Vinberg’s problem. Functional Analysis Appl. 16 (1982), 4, 311-313.Google Scholar
  20. [20]
    Polterovich L. The geometry of the group of symplectic diffeomorphisms. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001.Google Scholar
  21. [21]
    Polterovich L. Hofer’s diameter and Lagrangian intersections,. Int.Math.Res.Notices 4 (1998), 217-223.Google Scholar
  22. [22]
    Polterovich,L.;Siburg,K.F.;On the asymptotic geometry of area preserving maps .Math.Res.Lett. 7. 233-243 (2000).Google Scholar
  23. [23]
    Robbin, J.W. and Salamon, D.A. The Maslov index for paths, Topology, 32, (1993), 827-844.Google Scholar
  24. [24]
    Spivak,M. A Comprehensive Introduction to Differential Geometry, Volume 2, third edition, 1999, Publish or Perish,Inc.Google Scholar
  25. [25]
    Weinstein,A. Lectures on Symplectic manifolds, Regional Conference Series in Mathematics 29, Amer. Math. Soc., 1977.Google Scholar
  26. [26]
    Weinstein,A. Cohomology of Symplectomorphism Groups and Critical Values of Hamiltonians. Math.Zeit., 201, (1989), 75-82. .Google Scholar
  27. [27]
    Woodhouse,N.M.J. Geometric Quantization, second edition, Oxford University Press, 1991.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsETHZürichSwitzerland

Personalised recommendations