Flat bi-Hamiltonian structures and invariant densities

TL;DR

This paper characterizes flat bi-Hamiltonian structures on odd-dimensional manifolds, showing they are equivalent to the existence of a common invariant density preserved by all Hamiltonian vector fields of both structures.

Contribution

It establishes a necessary and sufficient condition for flatness of bi-Hamiltonian structures in terms of invariant densities, linking geometric flatness to dynamical invariants.

Findings

Flat bi-Hamiltonian structures are characterized by the existence of a common invariant density.

A generic bi-Hamiltonian structure is flat if and only if it admits a local density preserved by all Hamiltonian vector fields.

The result provides a geometric criterion for flatness in terms of invariant densities.

Abstract

A bi-Hamiltonian structure is a pair of Poisson structures $\mathcal P$, $\mathcal Q$ which are compatible, meaning that any linear combination $\alpha \mathcal P + \beta \mathcal Q$ is again a Poisson structure. A bi-Hamiltonian structure $(\mathcal P, \mathcal Q)$ is called flat if $\mathcal P$ and $\mathcal Q$ can be simultaneously brought to a constant form in a neighborhood of a generic point. We prove that a generic bi-Hamiltonian structure $(\mathcal P, \mathcal Q)$ on an odd-dimensional manifold is flat if and only if there exists a local density which is preserved by all vector fields Hamiltonian with respect to $\mathcal P$, as well as by all vector fields Hamiltonian with respect to $\mathcal Q$.