Hamiltonian Structures of Reductions of the Benney System

We show how to construct the Hamiltonian structures of any reduction of the Benney chain (dispersionless KP). The construction follows the scheme suggested by Ferapontov, leading in general to nonlocal Hamiltonian structures. In some special cases these reduce to local structures. All the geometric objects which define the Poisson bracket, the metric, connection and Riemann curvature, are obtained explicitly, in terms of the n-parameter family of conformal maps associated with the reduction.