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

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Abstract

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.

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Authors and Affiliations

  1. Department de Matemàtica I, Universitat Politecnica de Catalunya, 08028, Barcelona, Spain

    Yuri N. Fedorov

  2. Mathematical Institute SANU, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000, Belgrad, Serbia

    Božidar Jovanović

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  1. Yuri N. Fedorov
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  2. Božidar Jovanović
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Correspondence to Yuri N. Fedorov.

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Fedorov, Y.N., Jovanović, B. Geodesic flows and Neumann systems on Stiefel varieties: geometry and integrability. Math. Z. 270, 659–698 (2012). https://doi.org/10.1007/s00209-010-0818-y

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  • DOI: https://doi.org/10.1007/s00209-010-0818-y

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