Mathematische Zeitschrift

, Volume 270, Issue 3, pp 659–698

Geodesic flows and Neumann systems on Stiefel varieties: geometry and integrability

Authors

    • Department de Matemàtica IUniversitat Politecnica de Catalunya
  • Božidar Jovanović
    • Mathematical Institute SANUSerbian Academy of Sciences and Arts
Article

DOI: 10.1007/s00209-010-0818-y

Cite this article as:
Fedorov, Y.N. & Jovanović, B. Math. Z. (2012) 270: 659. doi:10.1007/s00209-010-0818-y

Abstract

We study integrable geodesic flows on Stiefel varieties Vn,r = SO(n)/SO(nr) given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics. We also consider natural generalizations of the Neumann systems on Vn,r with the above metrics and proves their integrability in the non-commutative sense by presenting compatible Poisson brackets on (T*Vn,r)/SO(r). Various reductions of the latter systems are described, in particular, the generalized Neumann system on an oriented Grassmannian Gn,r and on a sphere Sn−1 in presence of Yang–Mills fields or a magnetic monopole field. Apart from the known Lax pair for generalized Neumann systems, an alternative (dual) Lax pair is presented, which enables one to formulate a generalization of the Chasles theorem relating the trajectories of the systems and common linear spaces tangent to confocal quadrics. Additionally, several extensions are considered: the generalized Neumann system on the complex Stiefel variety Wn,r = U(n)/U(nr), the matrix analogs of the double and coupled Neumann systems.

Mathematics Subject Classification (2000)

17B8053D2570H0670H3370H45
Download to read the full article text

Copyright information

© Springer-Verlag 2010