Flatness of generic Poisson pairs in odd dimension

Abstract

Given a $(m-2)$-form $\zw$ and a volume form $\zW$ on a $m$-manifold one defines a bi-vector $\zL$ by setting $\zL(\za,\zb)={\frac {\za\zex\zb\zex\zw} {\zW}}$ for any $1$-forms $\za,\zb$. In this way, locally, a Poisson pair, or bi-Hamiltonian structure, $(\zL,\zL_1 )$ is always represented by a couple of $(m-2)$-forms $\zw,\zw_1$ and a volume form $\zW$. Here one shows that, for $m\zmai 5$ and odd and $(\zL,\zL_1 )$ generic, $(\zL,\zL_1 )$ is flat if and only if there exists a $1$-form $\zl$ such that $d\zw=\zl\zex\zw$ and $d\zw_1 =\zl\zex\zw_1$.