Mathematische Zeitschrift

, Volume 270, Issue 3–4, pp 659–698 | Cite as

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

Article

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)

17B80 53D25 70H06 70H33 70H45 

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department de Matemàtica IUniversitat Politecnica de CatalunyaBarcelonaSpain
  2. 2.Mathematical Institute SANUSerbian Academy of Sciences and ArtsBelgradSerbia

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