Nijenhuis geometry, multihamiltonian systems of hydrodynamic type and geodesic equivalence

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 $\le n+2$ 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.

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

differential geometry

Audience: researchers in the topic

---

Differential Geometry Seminar Torino

Series comments: This is a hybrid seminar organized by the Differential Geometry groups of Università and Politecnico di Torino.

To subscribe to our mailing list and receive announcements of the talks, please send an e-mail to dgseminar.torino@gmail.com

---