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

Alexey V Bolsinov,Vladimir S Matveev,Andrey Yu Konyaev

doi:10.1088/1361-6544/abed39

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.

Highlights

This paper continues the Nijenhuis geometry programme started in [3] and further developed in [4, 5, 24]
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
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

Summary

Introduction

This paper continues the Nijenhuis geometry programme started in [3] and further developed in [4, 5, 24]. This programme was initially motivated by the fact that Niejnhuis operators (i.e. fields of endomorphisms L = (Lij) with vanishing Nijenhuis torsion [31]) naturally appear in a number of different areas of geometry, algebra and mathematical physics For this reason their normal forms, singularities and global properties deserve more systematic study than before. We demonstrate such an overlap between geodesically equivalent pseudo-Riemannian metrics and compatible Poisson brackets of hydrodynamic type (see theorems 1–3 below). Once such a relationship is established and understood, one may try to transfer insights from one area to the other. Since this description is explicit, it can be used for analysis of singularities these operators may have

Basic definitions and main results

Proof of theorem 1

Proof of theorem 2

Proof of theorem 5