Skip to main content
SpringerLink
Log in

Complete involutive algebras of functions on cotangent bundles of homogeneous spaces

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract.

Homogeneous spaces of all compact Lie groups admit Riemannian metrics with completely integrable geodesic flows by means of C ∞–smooth integrals [9, 10]. The purpose of this paper is to give some constructions of complete involutive algebras of analytic functions, polynomial in velocities, on the (co)tangent bundles of homogeneous spaces of compact Lie groups. This allows us to obtain new integrable Riemannian and sub-Riemannian geodesic flows on various homogeneous spaces, such as Stiefel manifolds, flag manifolds and orbits of the adjoint actions of compact Lie groups.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aloff, S., Wallach, R.: An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures. Bull. Am. Math. Soc. 81 (1), 93–97 (1975)

    MATH  Google Scholar 

  2. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics. Dynamical Systems III, Springer, 1988

  3. Babenko, I.K., Nekhoroshev, N.N.: Complex structures on two-dimensional tori that admit metrics with a nontrivial quadratic integral. Mat. Zametki 58 (5), 643–652 (Russian) (1995)

    MATH  Google Scholar 

  4. Bazaikin, Ya.V.: Double quotients of Lie groups with an integrable geodesic flows. Sibirsk. Mat. Zh. 41 (3), 513–530 (2000) (Russian); English translation: Siberian Math. J. 41 (3), 419–432 (2000)

    Google Scholar 

  5. Bolsinov, A.V.: Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution. Izv. Acad. Nauk SSSR, Ser. matem. 55(1), 68–92 (1991) (Russian); English translation: Math. USSR-Izv. 38 (1), 69–90 (1992)

    Google Scholar 

  6. Bolsinov, A.V., Kozlov, V.V., Fomenko, A.T.: The Maupertuis’s principle and geodesic flows on S 2 arising from integrable cases in the dynamics of a rigid body. Uspekhi Mat. Nauk 50 (3), 3–32 (1995) (Russian); English translation: Russ. Math. Surv. 50, 473–501 (1995)

    MATH  Google Scholar 

  7. Bolsinov, A.V., Taimanov, I.A.: Integrable geodesic flow with positive topological entropy. Invent. math. 140, 639–650 (2000); arXiv: math.DG/9905078

    Article  MathSciNet  MATH  Google Scholar 

  8. Bolsinov, A.V., Taimanov, I.A.: Integrable geodesic flows on the suspensions of toric automorphisms. Tr. Mat. Inst. Steklova 231, 46–63 (2000) (Russian); English translation: Proc. Steklov. Inst. Math. 231 (4), 42–58 (2000); arXiv: math.DG/9911193

    MathSciNet  MATH  Google Scholar 

  9. Bolsinov, A.V., Jovanović, B.: Integrable geodesic flows on homogeneous spaces. Matem. Sbornik 192 (7), 21–40 (2001) (Russian): English translation: Sb. Mat. 192 (7–8), 951–969 (2001)

    MATH  Google Scholar 

  10. Bolsinov, A.V., Jovanović, B.: Non-commutative integrability, moment map and geodesic flows. Ann. Global Anal. Geom. 23 (4), 305–322 (2003); arXiv: math-ph/ 0109031

    Article  MATH  Google Scholar 

  11. Bordemann, M.: Hamiltonsche Mechanik auf homogenen Räumen. Diplomarbeit, Fakultät für Physik, Universität Freiburg, May 1985

  12. Brailov, A.V.: Construction of complete integrable geodesic flows on compact symmetric spaces. Izv. Acad. Nauk SSSR, Ser. matem. 50 (2), 661–674 (1986) (Rfussian); English translation: Math. USSR-Izv

    Google Scholar 

  13. Butler, L.: A new class of homogeneous manifolds with Liouville-integrable geodesic flows. C. R. Math. Acad. Sci. Soc. R. Can. 21 (4), 127–131 (1999)

    MATH  Google Scholar 

  14. Guillemin, V., Sternberg, S.: On collective complete integrability according to the method of Thimm. Ergod. Th. & Dynam. Sys. 3, 219–230 (1983)

    Google Scholar 

  15. Guillemin, V., Sternberg, S.: Symplectic techniques in Physics. Cambridge University Press, 1984

  16. Jovanović, B.: Geometry and integrability of Euler–Poincaré-Suslov equations. Nonlinearity 14 (6), 1555–1657 (2001); arXiv: math-ph/0107024

    Article  Google Scholar 

  17. Kiyohara, K.: Two-dimensional geodesic flows having first integrals of higher degree. Math. Ann. 320, 487–505 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Kruglikov, B.: Examples of integrable sub-Riemannian geodesic flows. J. Dynam. Control Systems. 8 (3), 323–340 (2002); arXiv: math.DS/0105128

    Article  MATH  Google Scholar 

  19. Kolokol’tsov, V.N.: Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities. Izv. Akad. Nauk SSSR Ser. Mat. 46 (5), 994–1010 (1982) (Russian); English translation: Math. USSR Izv. 21, 291–306 (1983)

    MATH  Google Scholar 

  20. Kramer, M.: Sphärische Untergruppen in kompakten zusammenhangenden Liegruppen. Compositio Math. 38, 129–153 (1979)

    MathSciNet  Google Scholar 

  21. Mishchenko, A.S.: Integration of geodesic flows on symmetric spaces. Mat. zametki 31 (2), 257–262 (1982) (Russian); English translation: Math. Notes. 31 (1–2), 132–134 (1982)

    MATH  Google Scholar 

  22. Mishchenko, A.S., Fomenko, A.T.: Euler equations on finite-dimensional Lie groups. Izv. Acad. Nauk SSSR, Ser. matem. 42 (2), 396–415 (1978) (Russian); English translation: Math. USSR-Izv. 12 (2), 371–389 (1978)

    Google Scholar 

  23. Mishchenko, A.S., Fomenko, A.T.: Generalized Liouville method of integration of Hamiltonian systems. Funkts. Anal. Prilozh. 12 (2), 46–56 (1978) (Russian); English translation: Funct. Anal. Appl. 12, 113–121 (1978)

    MATH  Google Scholar 

  24. Mikityuk, I.V.: Homogeneous spaces with integrable G–invariant Hamiltonian flows. Izv. Acad. Nauk SSSR, Ser. Mat. 47 (6), 1248–1262 (1983) (Russian); English translation in Math. USSR-Izv.

    Google Scholar 

  25. Mikityuk, I.V.: Integrability of the Euler equations associated with filtrations of semisimple Lie algebras. Matem. Sbornik 125 (4), 167 (1984) (Russian); English translation: Math. USSR Sbornik 53 (2), 541–549 (1986)

    MATH  Google Scholar 

  26. Mykytyuk, I.V., Stepin, A.M.: Classification of almost spherical pairs of compact simple Lie groups. In: Poisson geometry, Banach Center Publ., 51, Polish Acad. Sci., Warsaw, 2000, pp. 231–241

  27. Mykytyuk, I.V.: Actions of Borel subgroups on homogeneous spaces of reductive complex Lie groups and integrability. Composito Math. 127, 55–67 (2001)

    Article  MATH  Google Scholar 

  28. Panyushev, D.I.: Complexity of quasiaffine homogeneous varieties, t–decompositions, and affine homogeneous spaces of complexity 1, Adv. in Soviet Math. 8, 151–166 (1992)

    MATH  Google Scholar 

  29. Paternain, G.P., Spatzier, R.J.: New examples of manifolds with completely integrable geodesic flows. Adv. in Math. 108, 346–366 (1994); arXiv: math.DS/9201276

    Article  MathSciNet  MATH  Google Scholar 

  30. Selivanova, E.N.: New examples of integrable conservative systems on S 2 and the case of Goryachev-Chaplygin. Commun. Math. Phys. 207, 641–663 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Selivanova, E.N.: New families of conservative systems on S 2 possessing an integral of fourth degree in momenta. Ann. Global Anal. Geom. Math. 17, 201–219 (1999)

    Article  MATH  Google Scholar 

  32. Strichartz, R.S.: Sub-Riemannian geometry. J. Diff. Geometry, 24, 221–263 (1986); 30, 595–596 (1989)

    Google Scholar 

  33. Taimanov, I.A.: Topological obstructions to integrability of geodesic flows on non-simply-connected manifolds. Izv. Acad. Nauk SSSR, Ser. matem. 51 (2), 429–435 (1987) (Russian); English translation: Math. USSR-Izv., 30 (2), 403–409 (1988)

    Google Scholar 

  34. Taimanov, I.A.: Integrable geodesic flows of nonholonomic metrics. J. Dynamical and Control Systems, 3 (1), 129–147 (1997); arXiv: dg-ga/9610012

    Google Scholar 

  35. Thimm, A.: Integrable geodesic flows on homogeneous spaces. Ergod. Th. & Dynam. Sys. 1, 495–517 (1981)

    Google Scholar 

  36. Trofimov, V.V.: Euler equations on Borel subalgebras of semisimple Lie groups. Izv. Acad. Nauk SSSR, Ser. matem., 43 (3), 714–732 (1979) (Russian). English translation: Math. USSR-Izv.

    Google Scholar 

  37. Trofimov, V.V., Fomenko, A.T.: Algebra and geometry of integrable Hamiltonian differential equations. Moscow, Faktorial, 1995 (Russian)

  38. Vinberg, E.B.: Commutative homogeneous spaces and co-isotropic symplectic actions. Uspekhi Mat. Nauk 56 (1), 3–62 (2001) (Russian); English translation: Russian Math. Surveys 56 (1), 1–60 (2001)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics and Mathematics, 119899, Moscow, Russia

    Alexey V. Bolsinov

  2. Matematički Institut SANU, Kneza Mihaila 35, 11000, Beograd, Serbia, Yugoslavia

    Božidar Jovanović

Authors
  1. Alexey V. Bolsinov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Božidar Jovanović
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexey V. Bolsinov.

Additional information

Mathematics Subject Classification (2000): 70H06, 37J35, 53D17, 53D25

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bolsinov, A., Jovanović, B. Complete involutive algebras of functions on cotangent bundles of homogeneous spaces. Math. Z. 246, 213–236 (2004). https://doi.org/10.1007/s00209-003-0596-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-003-0596-x

Keywords

Search

Navigation