Authors: Fedorov, Yuri
Jovanović, Božidar 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Geodesic flows and Neumann systems on Stiefel varieties: Geometry and integrability
Journal: Mathematische Zeitschrift
Volume: 270
Issue: 3-4
First page: 659
Last page: 698
Issue Date: 1-Apr-2012
Rank: M21
ISSN: 0025-5874
DOI: 10.1007/s00209-010-0818-y
We study integrable geodesic flows on Stiefel varieties V n,r = SO(n)/SO(n-r) given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics. We also consider natural generalizations of the Neumann systems on V n,r with the above metrics and proves their integrability in the non-commutative sense by presenting compatible Poisson brackets on (T *V n,r)/SO(r). Various reductions of the latter systems are described, in particular, the generalized Neumann system on an oriented Grassmannian G n,r and on a sphere S n-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 W n,r = U(n)/U(n-r), the matrix analogs of the double and coupled Neumann systems.
Publisher: Springer Link

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