# Nijenhuis geometry, multihamiltonian systems of hydrodynamic type and geodesic equivalence

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Vladimir Matveev

created by raffero on 29 Dec 2020

modified on 10 Feb 2021

23 feb 2021
-- 17:00

Differential Geometry Seminar Torino (online)

**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 $\leq n+2$ 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.

The talk based on a series of joint publications with A. Bolsinov (Lboro) and A. Konyaev (Moscow); the most related one is https:/arxiv.org*abs*2009.07802

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The link of the virtual room will be sent one day before the talk to all registered participants. To register, please send an e-mail to dgseminar.torino@gmail.com