Nijenhuis geometry, multihamiltonian systems of hidrodynamic type and geodesic equivalence
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 less than or equal to 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 is 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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