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 $$V_{3,2}\cong\mathrm{SO}(3)$$ , 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 $$V_{n,r}$$ .