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

Abstract

We study integrable geodesic flows on Stiefel varieties Vn,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 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(n − r ), the matrix analogs of the double and coupled Neumann systems.