casimirs_and_commuting_flows_Nonlinearity.pdf (351.45 kB)

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

preprint
posted on 02.10.2020, 09:30 by Alexey BolsinovAlexey Bolsinov, Andrey Yu Konyaev, Vladimir S Matveev
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.

• Science

Department

• Mathematical Sciences

arXiv

arXiv

Version

SMUR (Submitted Manuscript Under Review)

2020-09-16

arXiv:2009.07802

en

Depositor

Dr Alexey Bolsinov. Deposit date: 30 September 2020

Exports

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