Reciprocal transformations and local Hamiltonian structures of hydrodynamic-type systems

We start from a hyperbolic Dubrovin and Novikov (DN) hydrodynamic-type system of dimension n which possesses Riemann invariants and we settle the necessary conditions on the conservation laws in the reciprocal transformation so that, after such a transformation of the independent variables, one of the metrics associated with the initial system is flat. We prove the following statement: let n be greater or equal to 3 in the case of reciprocal transformations of a single independent variable or n be greater or equal to 5 in the case of transformations of both the independent variables; then the reciprocal metric may be flat only if the conservation laws in the transformation are linear combinations of the canonical densities of conservation laws, i.e. the Casimirs, the momentum and the Hamiltonian densities associated with the Hamiltonian operator for the initial metric. Then, we restrict ourselves to the case in which the initial metric is either flat or of constant curvature and we classify the reciprocal transformations of one or both the independent variables so that the reciprocal metric is flat. Such characterization has an interesting geometric interpretation: the hypersurfaces of two diagonalizable DN systems of dimension n greater than 5 are Lie equivalent if and only if the corresponding local Hamiltonian structures are related by a canonical reciprocal transformation.