Geodesic Flows and Neumann Systems on Stiefel Varieties. Geometry and Integrability

TL;DR

This paper investigates integrable geodesic flows and Neumann systems on Stiefel varieties, establishing their integrability, exploring reductions, and presenting new Lax pairs and geometric relations.

Contribution

It introduces new integrable systems on Stiefel varieties, proves their non-commutative integrability, and provides alternative Lax pairs and geometric characterizations.

Findings

Proved integrability of geodesic flows on Stiefel varieties.

Developed compatible Poisson brackets for Neumann systems.

Presented a dual Lax pair and generalized Chasles theorem.

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.