Integrable geodesic flows on homogeneous spaces

Abstract

AbstractA method is exposed which allows the construction of families of first integrals in involution for Hamiltonian systems which are invariant under the Hamiltonian action of a Lie group G. This is applied to invariant Hamiltonian systems on the tangent bundles of certain homogeneous spaces M = G/K. It is proved, for example, that every such invariant Hamiltonian system is completely integrable if M is a real or complex Grassmannian manifold or SU(n + 1)/SO(n + 1) or a distance sphere in ℂPn+1. In particular, the geodesic flows of these homogeneous spaces are integrable.