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)



Singular cosphere bundle reduction

Authors: Oana M. Dragulete, Tudor S. Ratiu and Miguel Rodríguez-Olmos
Journal: Trans. Amer. Math. Soc. 359 (2007), 4209-4235
MSC (2000): Primary 53D10, 53D20
Published electronically: April 11, 2007
MathSciNet review: 2309182
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies singular contact reduction for cosphere bundles at the zero value of the momentum map. A stratification of the singular quotient, finer than the contact one and better adapted to the bundle structure of the problem, is obtained. The strata of this new stratification are a collection of cosphere bundles and coisotropic or Legendrian submanifolds of their corresponding contact components.

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

  • 1. C. Albert, Le théorème de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact, J. Geom. Physics, 6 (1989), 627-649. MR 1076705 (91k:58033)
  • 2. D.E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Math. 203, Birkhäuser, Boston, Basel, 2002. MR 1874240 (2002m:53120)
  • 3. R.H. Cushman and L.M. Bates, Global Aspects of Classical Integrable Systems, Birkhäuser, Basel, 1997. MR 1438060 (98a:58083)
  • 4. O. Dragulete, L. Ornea, Non-zero Contact and Sasakian Reduction, Differential Geom. Appl. 24 (2006), 260-270. MR 2216940
  • 5. O. Dragulete, L. Ornea, T.S. Ratiu, Cosphere bundle reduction in contact geometry, J. Symplectic Geom., 1 (2003), 695-714. MR 2039161 (2004m:53141)
  • 6. J. J. Duistermaat, J. A. Kolk, Lie Groups, Universitext, Springer-Verlag, 2000. MR 1738431 (2001j:22008)
  • 7. T. Ekholm, J. B. Etnyre, Invariants of Knots, Embeddings and Immersions via Contact Geometry, math.GT/0412517. MR 2189927
  • 8. H. Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc., 121 (1997), 455-464. MR 1434654 (98f:53027)
  • 9. H. Geiges, A Brief History of Contact Geometry and Topology, Expo. Math., 19 (2001), 25-53. MR 1820126 (2002c:53129)
  • 10. V. Guillemin and S. Sternberg, Homogeneous quantization and multiplicities of group representations, J. Funct. Anal., 47 (1982), 344-380. MR 0665022 (84d:58034)
  • 11. E. Lerman, Contact cuts, Israel J. Math., 124 (2001), 77-92. MR 1856505 (2002g:53156)
  • 12. E. Lerman, C. Willett, The topological structure of contact and symplectic quotients, Internat. Math. Res. Notices, 1 (2001), 33-52. MR 1809496 (2001j:53112)
  • 13. F. Loose, Reduction in contact geometry, J. Lie Theory, 11, no 1. (2001), 9-22. MR 1828281 (2002g:53147)
  • 14. J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry, Springer Texts in Appl. Math. 17, Second edition, Second printing 2003. MR 1304682 (95i:58073)
  • 15. J-P. Ortega and T.R. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, Volume 222, Birkhäuser, Boston, 2004. MR 2021152 (2005a:53144)
  • 16. R.S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2), 73 (1961), 295-323. MR 0126506 (23:A3802)
  • 17. M. Perlmutter, M. Rodríguez-Olmos, M. E. Sousa-Dias, On the geometry of reduced cotangent bundles at zero momentum, math.SG/0310437.
  • 18. M. J. Pflaum, Analytic and Geometric Study of Stratified Spaces, Lecture Notes in Mathematics, volume 510, Springer Verlag, 2001. MR 1869601 (2002m:58007)
  • 19. T. Ratiu, R. Schmid, The differentiable structure of three remarkable diffeomorphism groups, Math. Z., 177 (1981), 81-100. MR 0611471 (83d:57024)
  • 20. C. Willett, Contact reduction, Trans. Amer. Math. Soc., 354 (2002), 4245-4260. MR 1926873 (2003m:53152)
  • 21. M. Zambon, C. Zhu, Contact reduction and groupoid actions, math.DG/0405047.

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

Oana M. Dragulete
Affiliation: Section de mathématiques, EPFL, CH-1015 Lausanne, Switzerland and Department of Mathematics, University “Politehnica” of Bucharest, Romania

Tudor S. Ratiu
Affiliation: Section de mathématiques, EPFL, CH-1015 Lausanne, Switzerland

Miguel Rodríguez-Olmos
Affiliation: Section de mathématiques, EPFL, CH-1015 Lausanne, Switzerland

Keywords: Contact manifold, cotangent, cosphere bundle, momentum map, singular reduction.
Received by editor(s): June 30, 2005
Published electronically: April 11, 2007
Additional Notes: The first and the second authors thank the Swiss National Science Foundation for partial support
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society