Hostname: page-component-74656f9884-rdgjk Total loading time: 0 Render date: 2026-11-08T14:35:37.813Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrable geodesic flows on homogeneous spaces

Published online by Cambridge University Press:  19 September 2008

A. Thimm
A. Thimm
Affiliation:
Mathematisches Institut der Universität Bonn, Wegelerstrasse 10, D5300 Bonn 1, West Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

A method is exposed which allows the construction of families of first integrals in involution for Hamiltonian systems which are invariant under the Hamiltonian action of a Lie group G. This is applied to invariant Hamiltonian systems on the tangent bundles of certain homogeneous spaces M = G/K. It is proved, for example, that every such invariant Hamiltonian system is completely integrable if M is a real or complex Grassmannian manifold or SU(n + 1)/SO(n + 1) or a distance sphere in ℂPn+1. In particular, the geodesic flows of these homogeneous spaces are integrable.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 1 , Issue 4 , December 1981 , pp. 495 - 517
Copyright
Copyright © Cambridge University Press 1981

References

REFERENCES

[1]Arnold, V. I.. Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics 60. Springer-Verlag: Berlin, 1978.CrossRefGoogle Scholar
[2]Guillemin, V. & Sternberg, S.. The moment map and collective motion. Ann. Phys. 127 (1980), 220–253.CrossRefGoogle Scholar
[3]Helgason, S.. Differential Geometry and Symmetric Spaces. Academic Press: New York, 1962.Google Scholar
[4]Heigason, S.. Fundamental solutions of invariant differential operators on symmetric spaces. Amer. J. Math. 86 (1954) 565–601.CrossRefGoogle Scholar
[5]Kobayashi, S. & Nomizu, K.. Foundations of Differential Geometry II. Wiley (Interscience): New York, 1969.Google Scholar
[6]Mishchenko, A. S. & Fomenko, A. T.. Euler equations on finite-dimensional Lie groups. Izv. Akad. Nauk SSSR, Ser Mat. 42 (1978), 396–415.Google Scholar
[7]Moser, J.. Various aspects of integrable Hamiltonian systems. In Proc. CIME Conference,Bressanone, Italy1978. Prog. Math. 8, Birkhauser (1980).Google Scholar
[8]Steinberg, R.. Invariants of finite reflection groups. Canad. J. Math. 12 (1960), 616–618.CrossRefGoogle Scholar
[9]Thimm, A., Integrabilität beim geodätischen Fluβ. Bonner Math. Schriften 103 (1978).Google Scholar
[10]Thimm, A.. Über die Integrabilität geodatischer Flüsse auf homogenen Räumen. Dissertation: Bonn, 1980.Google Scholar
[11]Ziller, W.. The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces. Comment. Math. Helv. 52 (1977), 573–590.CrossRefGoogle Scholar