On Bi-hamiltonian Flows and Their Realizations as Curves in Real Semisimple Homogenous Manifolds

In this paper we describe a reduction process that allows us to define Hamiltonian structures on the manifold of differential invariants of parametrized curves for any homogeneous manifold of the form G/H, with G semisimple. We also prove that equations that are Hamiltonian with respect to the first of these reduced brackets automatically have a geometric realization as an invariant flow of curves in G/H. This result applies to some well-known completely integrable systems. We study in detail the Hamiltonian structures associated to the sphere SO(n + 1)/SO(n).