A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderno S/19, Dipartimento di Matematica dell'Universita' di Milano

In this report (172 pages, with 4 appendices) we introduce the concept of Posson-Nijenhuis (PN) manifold and we show that the integrable Hamiltonian systems are the "fundamental fields" of such manifolds. In particular, we expl;icitly construct a simple model of infinite-dimensional PN manifold and we show that it gives rise to the Gelfand-Dickey equations. The many advantages of the present approach seem to be its conceptual simplicity and the property of being systematic. The different equations are obtained as different reductions of a single PN structure, performed over special submanifolds picked out by the geometry of the PN manifold itself.