Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type

TL;DR

This paper links geodesically equivalent metrics with compatible Poisson brackets of hydrodynamic type, constructing maximal families of such structures and generalizing classical results to Einstein metrics.

Contribution

It establishes a connection between geodesic equivalence and Poisson brackets, constructs maximal compatible families, and generalizes Sinjukov's result to Einstein metrics.

Findings

Constructed a maximal family of compatible Poisson structures of dimension (n+1)(n+2)/2.

Proved the uniqueness of polynomial families of compatible Poisson structures of dimension n+2.

Generalized Sinjukov's result from constant curvature to Einstein metrics.

Abstract

We connect two a priori unrelated topics, theory of geodesically equivalent metrics in differential geometry, and 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 $\le n+2$ 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 generalise a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.