Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type

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. In addition, we generalise a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.

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References in corpus (11)

• Integrable hydrodynamic chains
• Compatible Poisson brackets of hydrodynamic type
• Complete Einstein metrics are geodesically rigid
• Nijenhuis Geometry
• Nijenhuis geometry II: Left-symmetric algebras and linearization problem for Nijenhuis operators
• Local normal forms for geodesically equivalent pseudo-Riemannian metrics
• $F$-manifolds, multi-flat structures and Painlevé transcendents
• Projectively related metrics, Weyl nullity, and metric projectively invariant equations
• Quasilinear systems of Jordan block type and the mKP hierarchy
• Applications of Nijenhuis geometry: Nondegenerate singular points of Poisson-Nijenhuis structures
• Metric connections in projective differential geometry

Cited by in corpus (4)

• Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures
• Orthogonal separation of variables for spaces of constant curvature
• Proof of Laugwitz Conjecture and Landsberg Unicorn Conjecture for Minkowski norms with $SO(k)\times SO(n-k)$-symmetry
• Integrating Nijenhuis Structures