Discrete Geodesic Flows on Stiefel Manifolds

Abstract

We study integrable discretizations of geodesic flows of Euclidean metrics on the cotangent bundles of the Stiefel manifolds Vn,r. In particular, for n=3 and r=2, after the identification V3,2≅SO(3), we obtain a discrete analog of the Euler case of the rigid body motion corresponding to the inertia operator I=(1,1,2). In addition, billiard-type mappings are considered; one of them turns out to be the “square root” of the discrete Neumann system on Vn,r.