Singular Manakov flows and geodesic flows on homogeneous spaces of SO(N)

Abstract

We prove complete integrability of the Manakov-type SO(n)-invariant geodesic flows on homogeneous spaces SO(n)/SO(k1) ×...× SO(kr), for any choice of k1,...,kr, k1 + ,..., + kr ≤ n. In particular, a new proof of the integrability of a Manakov symmetric rigid body motion around a fixed point is presented. Also, the proof of integrability of the SO(n)-invariant Einstein metrics on SO(k1 + k2 + k3)/SO(k1) × SO(k2) × SO(k3) and on the Stiefel manifolds V (n, k) = SO(n)/SO(k) is given.