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

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.