### Abstract

A bi-Hamiltonian structure is a pair of Poisson structures P, Q which are compatible, meaning that any linear combination αP+ βQ is again a Poisson structure. A bi-Hamiltonian structure (P, Q) is called flat if P and Q can be simultaneously brought to a constant form in a neighborhood of a generic point. We prove that a generic bi-Hamiltonian structure (P, 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 P, as well as by all vector fields Hamiltonian with respect to Q.

Original language | English (US) |
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Pages (from-to) | 1415-1427 |

Number of pages | 13 |

Journal | Letters in Mathematical Physics |

Volume | 106 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 1 2016 |

Externally published | Yes |

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### Keywords

- bi-Hamiltonian structures
- Casimir functions
- invariant densities

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics