Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Symplectic leaves of the Gel'fand-Dikii brackets and homotopy classes of nondegenerate curves

  • 69 Accesses

  • 19 Citations

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

Literature Cited

  1. 1.

    I. M. Gel'fand and L. A. Dikii, "Fractional powers of operator and Hamiltonian systems," Funkts. Anal. Prilozhen.,10, No. 4, 13–20 (1976).

  2. 2.

    I. M. Gel'fand and L. A. Dikii, The Family of the Hamiltonian Structures That are Connected with Integrable Nonlinear Equations [in Russian], Preprint, Inst. Prikl. Matematiki, No. 136 (1978).

  3. 3.

    J.-A. Little, "Nondegenerate homotopics of curves on the unit 2-sphere," J. Diff. Geom.,4, No. 3, 339–348 (1970).

  4. 4.

    B. Z. Shapiro and M. Z. Shapiro, "On the number of connected components in the space of closed nondegenerate curves on Sn," Submitted to Bull. Amer. Math. Soc. in 1984.

  5. 5.

    N. H. Kuiper, "Locally projective spaces of dimension one," Michigan Math. J.,2, No. 2, 95–97 (1953–54).

  6. 6.

    V. P. Lazutkin and T. F. Pankratova, "Normal forms and versal deformations for the Hille equations," Funkts. Anal. Prilozhen.,9, No. 4, 41–48 (1975).

  7. 7.

    A. A. Kirillov, "Infinite-dimensional Lie groups: their orbits, invariants and representations. Geometry of moments," in: H.-D. Doebner and T. D. Palev (ed.), Twistor Geometry and Nonlinear Systems, Lect. Notes in Math., Vol. 970, Springer Verlag, Berlin—New York (1982), pp. 101–123.

  8. 8.

    G. Segal, "Unitary representations of some infinite dimensional groups," Comm. Math. Phys.,80, No. 3, 301–342 (1981).

  9. 9.

    V. Yu. Ovsienko, "Classification of linear differential equations of third order and symplectic leaves of the Gel'fand—Dikii bracket," Mat. Zametki,47 (1980).

  10. 10.

    V. Yu. Ovsienko, O. D. Ovsienko, and Yu. V. Chekanov, "Classification of contact-projective structures on the supercircle," Usp. Mat. Nauk,44, No. 3, 167–168 (1989).

  11. 11.

    B. A. Khesin, "Invariants of the Hamiltonian kdV-structures," Usp. Mat. Nauk,45, No. 1 (1990).

  12. 12.

    A. B. Zamolodchikov, "Additional symmetries in the two-dimensional quantum conformal theory of field," Teor. Mat. Fiz.,65, No. 3, 347–359 (1985).

  13. 13.

    S. L. Luk'yanov, "Quantization of the Gel'fand—Dikii bracket," Funkts. Anal. Prilozhen.,22, No. 4, 1–10 (1988).

  14. 14.

    E. Witten, "Coadjoint orbits of the Virasoro group," Comm. Math. Phys.,114, No. 1, 1–53 (1988).

  15. 15.

    V. G. Drinfel'd and V. V. Sokolov, "Lie algebras and Korteweg—de Vries type equations," Itogi Nauki Tekhniki SSSR, Sovremennye Problemy Matematiki,24, 81–180 (1984).

  16. 16.

    V. I. Arnol'd, Supplementary Chapters of the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1978).

  17. 17.

    T. G. Khovanova, "The Gel'fand—Dikii Lie algebras and the Virasoro algebra," Funkts. Anal. Prilozhen.,20, No. 4, 89–90 (1986).

  18. 18.

    P. Hartman, "On nonoscillatory linear differential equations with monotonic coefficients," Am. J. Math.,76, 207–219 (1954).

  19. 19.

    A. M. Vershik and V. Ya. Gershkovich, "Nonholonomic dynamical systems, Geometry of distributions and variational problems," Itogi Nauki Tekhniki SSSR, Sovremennye Problemy Matematiki. Fundamental'nye Napravleniya,16, 5–226 (1987).

  20. 20.

    U. Hammenstädt, "Zur Theorie von Carnot—Caratheodory Metriken und ihren Anwendungen," Bonn. Math. Schr., No. 180, 1–64 (1987).

  21. 21.

    T. G. Khovanova, "The structure of a Lie superalgebra on eigenfunctions and jets of the resolvent kernel near the diagonal for differential operators of n-th order," Funkts. Anal. Prilozhen.,20, No. 2, 86–89 (1986).

  22. 22.

    B. Z. Shapiro, M. Z. Shapiro, and B. A. Khesin, "On the number of connected components of nondegenerate curves on S2 with given monodromy," Submitted to J. Diff. Geom. in 1986.

Download references

Additional information

M. V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 1, pp. 38–47, January–March, 1990.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ovsienko, V.Y., Khesin, B.A. Symplectic leaves of the Gel'fand-Dikii brackets and homotopy classes of nondegenerate curves. Funct Anal Its Appl 24, 33–40 (1990). https://doi.org/10.1007/BF01077916

Download citation


  • Functional Analysis
  • Homotopy Class
  • Symplectic Leave
  • Nondegenerate Curf