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 $(T^{\ast }{V}_{n,r})/SO(r)$. 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.

39 pages, to appear in Mathematische Zeitschrift

References in corpus (4)

• Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces
• Torus Actions and Integrable Systems
• A variational problem on Stiefel manifolds
• Singular Manakov Flows and Geodesic Flows on Homogeneous Spaces

Cited by in corpus (6)

• The Jacobi-Rosochatius problem on an ellipsoid: the Lax representations and billiards
• Heisenberg model in pseudo-Euclidean spaces
• Geodesic flows on Riemannian g.o. spaces
• Heisenberg model in pseudo-Euclidean spaces II
• Three natural mechanical systems on Stiefel varieties
• Contact magnetic geodesic and sub-Riemannian flows on $V_{n,2}$ and integrable cases of a heavy rigid body with a gyrostat