Bolsinov_2021_Nonlinearity_34_5136.pdf (695.38 kB)
Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type
journal contribution
posted on 2021-07-23, 14:30 authored by Alexey BolsinovAlexey Bolsinov, Andrey Konyaev, Vladimir MatveevWe 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 generalise a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.
History
School
- Science
Department
- Mathematical Sciences
Published in
NonlinearityVolume
34Issue
8Pages
5136-5162Publisher
IOP PublishingVersion
- AM (Accepted Manuscript)
Rights holder
© IOP Publishing Ltd & London Mathematical SocietyPublisher statement
This is an Open Access Article. It is published by IOP under the Creative Commons Attribution 3.0 Unported Licence (CC BY). Full details of this licence are available at: http://creativecommons.org/licenses/by/3.0/Acceptance date
2021-03-09Publication date
2021-06-28Copyright date
2021ISSN
0951-7715eISSN
1361-6544Publisher version
Language
- en
Depositor
Dr Alexey Bolsinov. Deposit date: 14 March 2021Usage metrics
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