Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 058, 22 pages      arXiv:1101.5587      https://doi.org/10.3842/SIGMA.2011.058
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on S2×S3

Charles P. Boyer
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA

Received January 28, 2011, in final form June 08, 2011; Published online June 15, 2011

Abstract
I begin by giving a general discussion of completely integrable Hamiltonian systems in the setting of contact geometry. We then pass to the particular case of toric contact structures on the manifold S2×S3. In particular we give a complete solution to the contact equivalence problem for a class of toric contact structures, Yp,q, discovered by physicists by showing that Yp,q and Yp',q' are inequivalent as contact structures if and only if p≠p'.

Key words: complete integrability; toric contact geometry; equivalent contact structures; orbifold Hirzebruch surface; contact homology; extremal Sasakian structures.

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References

  1. Abreu M., Kähler-Sasaki geometry of toric symplectic cones in action-angle coordinates, Port. Math. 67 (2010), 121-153, arXiv:0912.0492.
  2. Abreu M., Macarini L., Contact homology of good toric contact manifolds, arXiv:1005.3787.
  3. Arnold V.I., Mathematical methods of classical mechanics, Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York, 1978.
  4. Arnold V.I., Arnold's problems, Springer-Verlag, Berlin, 2004.
  5. Atiyah M.F., Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15.
  6. Audin M., Hamiltonian systems and their integrability, SMF/AMS Texts and Monographs, Vol. 15, American Mathematical Society, Providence, RI, 2008.
  7. Banyaga A., The structure of classical diffeomorphism groups, Mathematics and its Applications, Vol. 400, Kluwer Academic Publishers Group, Dordrecht, 1997.
  8. Banyaga A., The geometry surrounding the Arnold-Liouville theorem, in Advances in Geometry, Progr. Math., Vol. 172, Birkhäuser Boston, Boston, MA, 1999, 53-69.
  9. Banyaga A., Molino P., Géométrie des formes de contact complètement intégrables de type toriques, Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, 1991-1992 (Montpellier), Univ. Montpellier II, Montpellier, 1993, 1-25.
  10. Banyaga A., Molino P., Complete integrability in contact geometry, Preprint, The Pennsylvania State University, 1996, 25 pages.
  11. Blair D.E., Riemannian geometry of contact and symplectic manifolds, 2nd ed., Progr. Math., Vol. 203, Birkhäuser Boston, Inc., Boston, MA, 2010.
  12. Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor'kova N.G., Krasil'shchik I.S., Samokhin A.V., Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and conservation laws for differential equations of mathematical physics, Translations of Mathematical Monographs, Vol. 182, American Mathematical Society, Providence, RI, 1999.
  13. Boothby W.M., Wang H.C., On contact manifolds, Ann. of Math. (2) 68 (1958), 721-734.
  14. Boucetta M., Molino P., Géométrie globale des systèmes hamiltoniens complètement intégrables: fibrations lagrangiennes singulières et coordonnées action-angle à singularités, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 13, 421-424.
  15. Bourgeois F., A Morse-Bott approach to contact homology, Dissertation, Stanford University, 2002.
  16. Bourgeois F., A Morse-Bott approach to contact homology, in Symplectic and Contact Topology: Interactions and Perspectives (Toronto, Montreal, 2001), Fields Inst. Commun., Vol. 35, Amer. Math. Soc., Providence, RI, 2003, 55-77.
  17. Boyer C.P., The Sasakian geometry of the Heisenberg group, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52(100) (2009), 251-262, arXiv:0904.1406.
  18. Boyer C.P., Maximal tori in contactomorphism groups, arXiv:1003.1903.
  19. Boyer C.P., Extremal Sasakian metrics on S3-bundles over S2, Math. Res. Lett. 18 (2011), 181-189, arXiv:1002.1049.
  20. Boyer C.P., Galicki K., On Sasakian-Einstein geometry, Internat. J. Math. 11 (2000), 873-909, math.DG/9811098.
  21. Boyer C.P., Galicki K., A note on toric contact geometry, J. Geom. Phys. 35 (2000), 288-298, math.DG/9907043.
  22. Boyer C.P., Galicki K., Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.
  23. Boyer C.P., Galicki K., Simanca S.R., Canonical Sasakian metrics, Comm. Math. Phys. 279 (2008), 705-733, math.DG/0604325.
  24. Boyer C.P., Galicki K., Simanca S.R., The Sasaki cone and extremal Sasakian metrics, in Riemannian Topology and Geometric Structures on Manifolds, Progr. Math., Vol. 271, Birkhäuser Boston, Boston, MA, 2009, 263-290, arXiv:0801.0217.
  25. Boyer C.P., Pati J., On the equivalence problem for toric contact structures on S3-bundles over S2, work in progress.
  26. Bryant R.L., Bochner-Kähler metrics, J. Amer. Math. Soc. 14 (2001), 623-715, math.DG/0003099.
  27. Calabi E., Extremal Kähler metrics, in Seminar on Differential Geometry, Ann. of Math. Stud., Vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, 259-290.
  28. Cvetic M., Lü H., Page D.N., Pope C.N., New Einstein-Sasaki spaces in five and higher dimensions, Phys. Rev. Lett. 95 (2005), 071101, 4 pages, hep-th/0504225.
  29. David L., Gauduchon P., The Bochner-flat geometry of weighted projective spaces, in Perspectives in Riemannian geometry, CRM Proc. Lecture Notes, Vol. 40, Amer. Math. Soc., Providence, RI, 2006, 109-156.
  30. Delzant T., Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), 315-339.
  31. Duistermaat J.J., On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687-706.
  32. Eliashberg Y., Givental A., Hofer H., Introduction to symplectic field theory, Geom. Funct. Anal. (2000), special volume, 560-673, math.SG/0010059.
  33. Fassò F., Superintegrable Hamiltonian systems: geometry and perturbations, Acta Appl. Math. 87 (2005), 93-121.
  34. Friedrich T., Kath I., Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator, J. Differential Geom. 29 (1989), 263-279.
  35. Friedrich T., Kath I., 7-dimensional compact Riemannian manifolds with Killing spinors, Comm. Math. Phys. 133 (1990), 543-561.
  36. Gauntlett J.P., Martelli D., Sparks J., Waldram D., Sasaki-Einstein metrics on S2×S3, Adv. Theor. Math. Phys. 8 (2004), 711-734, hep-th/0403002.
  37. Gauntlett J.P., Martelli D., Sparks J., Waldram D., Supersymmetric AdS5 solutions of M-theory, Classical Quantum Gravity 21 (2004), 4335-4366, hep-th/0402153.
  38. Gauntlett J.P., Martelli D., Sparks J., Waldram D., Supersymmetric AdS backgrounds in string and M-theory, in AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries, IRMA Lect. Math. Theor. Phys., Vol. 8, Eur. Math. Soc., Zürich, 2005, 217-252, hep-th/0411194.
  39. Ghigi A., Kollár J., Kähler-Einstein metrics on orbifolds and Einstein metrics on spheres, Comment. Math. Helv. 82 (2007), 877-902, math.DG/0507289.
  40. Guillemin V., Sternberg S., Convexity properties of the moment mapping, Invent. Math. 67 (1982), 491-513.
  41. Hofer H., Polyfolds and a general Fredholm theory, arXiv:0809.3753.
  42. Karshon Y., Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc. 141 (1999), no. 672, 71 pages, dg-ga/9510004.
  43. Karshon Y., Maximal tori in the symplectomorphism groups of Hirzebruch surfaces, Math. Res. Lett. 10 (2003), 125-132, math.SG/0204347.
  44. Khesin B., Tabachnikov S., Contact complete integrability, Regul. Chaotic Dyn. 15 (2010), 504-520, arXiv:0910.0375.
  45. Legendre E., Existence and non existence of constant scalar curvature toric Sasaki metrics, Compos. Math., to appear, arXiv:1004.3461.
  46. Lerman E., Contact toric manifolds, J. Symplectic Geom. 1 (2002), 785-828, math.SG/0107201.
  47. Lerman E., Maximal tori in the contactomorphism groups of circle bundles over Hirzebruch surfaces, Math. Res. Lett. 10 (2003), 133-144, math.SG/0204334.
  48. Lerman E., Homotopy groups of K-contact toric manifolds, Trans. Amer. Math. Soc. 356 (2004), 4075-4083, math.SG/0204064.
  49. Libermann P., Legendre foliations on contact manifolds, Differential Geom. Appl. 1 (1991), 57-76.
  50. Libermann P., Marle Ch.-M., Symplectic geometry and analytical mechanics, Mathematics and its Applications, Vol. 35, D. Reidel Publishing Co., Dordrecht, 1987.
  51. Martelli D., Sparks J., Toric Sasaki-Einstein metrics on S2×S3, Phys. Lett. B 621 (2005), 208-212, hep-th/0505027.
  52. Martelli D., Sparks J., Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals, Comm. Math. Phys. 262 (2006), 51-89, hep-th/0411238.
  53. Mishchenko A.S., Fomenko A.T., A generalized Liouville method for the integration of Hamiltonian systems, Funct. Anal. Appl. 12 (1978), no. 2, 113-121.
  54. Nehorosev N.N., Action-angle variables, and their generalizations, Trudy Moskov. Mat. Obsc. 26 (1972), 181-198.
  55. Omori H., Infinite-dimensional Lie groups, Translations of Mathematical Monographs, Vol. 158, American Mathematical Society, Providence, RI, 1997.
  56. Pang M.-Y., The structure of Legendre foliations, Trans. Amer. Math. Soc. 320 (1990), 417-455.
  57. Tanno S., Geodesic flows on CL-manifolds and Einstein metrics on S3×S2, in Minimal Submanifolds and Geodesics (Proc. Japan-United States Sem., Tokyo, 1977), North-Holland, Amsterdam, 1979, 283-292.
  58. Tempesta P., Winternitz P., Harnad J., Miller W. Jr., Pogosyan G., Rodriguez M. (Editors), Superintegrability in classical and quantum systems (September 16-21, 2002, Montreal, Canada), CRM Proceedings and Lecture Notes, Vol. 37, American Mathematical Society, Providence, RI, 2004.

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