Discrete Geodesic Flows on Stiefel Manifolds

We study integrable discretizations of geodesic flows of Euclidean metrics on the cotangent bundles of the Stiefel manifolds V_n,r . In particular, for n=3 and r=2 , after the identification V_3,2≅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 V_n,r .