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We start from a hyperbolic DN hydrodynamic type system of dimension $n$ whichpossesses Riemann invariants and we settle the necessary conditions on theconservation laws in the reciprocal transformation so that, after such atransformation of the independent variables, one of the metrics associated tothe initial system be flat. We prove the following statement: let $n\ge 3$ inthe case of reciprocal transformations of a single independent variable or$n\ge 5$ in the case of transformations of both the independent variable; thenthe reciprocal metric may be flat only if the conservation laws in thetransformation are linear combinations of the canonical densities ofconservation laws, {\it i.e} the Casimirs, the momentum and the Hamiltoniandensities associated to the Hamiltonian operator for the initial metric. Then,we restrict ourselves to the case in which the initial metric is either flat orof constant curvature and we classify the reciprocal transformations of one orboth the independent variables so that the reciprocal metric is flat. Suchcharacterization has an interesting geometric interpretation: the hypersurfacesof two diagonalizable DN systems of dimension $n\ge 5$ are Lie equivalent ifand only if the corresponding local hamiltonian structures are related by acanonical reciprocal transformation.

Reciprocal transformations and local Hamiltonian structures of hydrodynamic type systems