Classical Neumann System on Stiefel Manifolds: Integrability, Geometric and Algebraic Aspects, and Linearization

Abstract

The Neumann system on a sphere is one of the basic classical examples of completely integrable systems. In this talk we give a review on the results concerning natural integrable generalizations of the Neumann systems to Stiefel manifolds [1,2,3,4]. Two Lax pairs for the systems are presented. A -matrix Lax representation enables us to prove non-commutative integrabilty of the Neumann systems, while a -matrix Lax representation implies a generalization of the Chasles theorem relating the trajectories of the systems and common linear spaces tangent to confocal quadrics. Also, by applying the second Lax representation we describe algebraic geometric properties of the systems. We show that generic complex invariant manifolds are open subsets of Prym varieties on which the flow is linear.