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 2020-10-02, 09:30 authored by Alexey BolsinovAlexey Bolsinov, Andrey Yu Konyaev, Vladimir S MatveevWe 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.
History
School
- Science
Department
- Mathematical Sciences
Published in
arXivPublisher
arXivVersion
- SMUR (Submitted Manuscript Under Review)
Rights holder
© The AuthorsPublication date
2020-09-16Notes
comments are welcomePublisher version
Other identifier
arXiv:2009.07802Language
- en