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

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.

Automated summary

The research connects the theory of geodesically equivalent metrics in differential geometry to the theory of compatible infinite-dimensional Poisson brackets of hydrodynamic type in mathematical physics. It demonstrates that a pair of geodesically equivalent metrics with one being flat produces a pair of such brackets. Through constructing Casimirs and corresponding commuting flows, the study describes two ways of producing a large family of compatible Poisson structures from a pair of geodesically equivalent metrics.