Applications of Nijenhuis geometry II: maximal pencils of multi-Hamiltonian structures of hydrodynamic type

Abstract

Abstract We connect two a priori unrelated topics, the theory of geodesically equivalent metrics in differential geometry, and the theory of compatible infinite-dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is (n + 1)(n + 2)/2 dimensional; we describe it completely and show that it is maximal. Another has dimension ⩽n + 2 and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension n + 2 is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalize a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.