COMPATIBLE METRICS AND THE DIAGONALIZABILITY OF NONLOCALLY BI-HAMILTONIAN SYSTEMS OF HYDRODYNAMIC TYPE

We study bi-Hamiltonian systems of hydrodynamic type with nonsingular (semisimple) nonlocal bi-Hamiltonian structures. We prove that all such systems of hydrodynamic type are diagonalizable and that the metrics of the bi-Hamiltonian structure completely determine the complete set of Riemann invariants constructed for any such system. Moreover, we prove that for an arbitrary nonsingular (semisimple) non-locally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all matrix differential-geometric objects related to this system, namely, the matrix (affinor) V-j(i)(u) of this system of hydrodynamic type, the metrics g(1)(ij)(u) and g(2)(ij)(u), the affinor v(j)(i)(u) = g(1)(is)(u)g(2,sj)(u), and also the affinors (w(1,n))(j)(i)(u) and (w(2,n))(j)(i)(u) of the nonsingular nonlocal bi-Hamiltonian structure of this system, are diagonal in these special "diagonalizing" local coordinates (Riemann invariants of the system). The proof is a natural corollary of the general results of our previously developed theories of compatible metrics and of nonlocal bi-Hamiltonian structures; we briefly review the necessary notions and results in those two theories.