| 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 \({{\mathcal Q}}\). |
