Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Contact reduction

Author: Christopher Willett
Journal: Trans. Amer. Math. Soc. 354 (2002), 4245-4260
MSC (2000): Primary 53D10, 53D20
Published electronically: May 23, 2002
MathSciNet review: 1926873
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this article I propose a new method for reducing co-oriented contact manifold $M$ equipped with an action of a Lie group $G$ by contact transformations. With a certain regularity and integrality assumption the contact quotient $M_\mu$ at $\mu \in \mathfrak{g}^*$ is a naturally co-oriented contact orbifold which is independent of the contact form used to represent the given contact structure. Removing the regularity and integrality assumptions and replacing them with one concerning the existence of a slice, which is satisfied for compact symmetry groups, results in a contact stratified space; i.e., a stratified space equipped with a line bundle which, when restricted to each stratum, defines a co-oriented contact structure. This extends the previous work of the author and E. Lerman.

References [Enhancements On Off] (What's this?)

  • [AM] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd Ed., Addison-Wesley, 1978. MR 81e:58025
  • [Al] C. Albert, Le théorème de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact, J. Geom. Phys. 6 (1989), no. 4, 627-649. MR 91k:58033
  • [BL] L. Bates and E. Lerman, Proper group actions and symplectic stratified spaces, Pac. J. of Math. (2) 181 (1997), 201-229. MR 98i:58085
  • [Ge] H. Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 3, 455-464. MR 98f:53027
  • [GS1] V. Guillemin and S. Sternberg, Homogeneous quantization and multiplicities of group representations, J. Funct. Anal. 47 (1982), no. 3, 344-380. MR 84d:58034
  • [GS2] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984. MR 86f:58054
  • [Ku] M. Kummer, On the construction of the reduced phase spaces of a Hamiltonian system with symmetry, Indiana Univ. Math. J. 30 (1981), no. 2, 281-291. MR 82e:58041
  • [L1] E. Lerman, Contact Cuts, Israel J. Math 124 (2001), 77-92. See
  • [L2] E. Lerman, Geodesic flows and contact toric manifolds, CRM Summer School in the Symplectic Geometry of Integrable Hamiltonian System, Notes. See
  • [LMTW] E. Lerman, E. Meinrenken, S. Tolman, and C. Woodward, Non-abelian convexity by symplectic cuts, Topology 37 (1998), no. 2, 245-259. MR 99a:58069
  • [LW] E. Lerman and C. Willett, The topological structure of contact and symplectic quotients, Internat. Math. Res. Notices (2001), no. 1, 33-52. MR 2001j:53112
  • [Lo] F. Loose, Reduction in contact geometry, J. Lie Theory 11 (2001), no. 1, 9-22. CMP 2001:12
  • [P] R. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. Math. 73 (1961), no. 2, 295-322. MR 23;A3802

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53D10, 53D20

Retrieve articles in all journals with MSC (2000): 53D10, 53D20

Additional Information

Christopher Willett
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Received by editor(s): November 21, 2001
Published electronically: May 23, 2002
Additional Notes: The author was supported by a National Science Foundation graduate Vertical Integration of Research and Education fellowship and the American Institute of Mathematics
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society