### Abstract

A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if the associated Veronese web is trivial. We suggest a simpler obstruction to flatness, which we call the curvature form of a Poisson pencil. This form can be defined in two ways: either via the Blaschke curvature form of the associated web, or via the Ricci tensor of a connection compatible with the pencil.We show that the curvature form of a Poisson pencil can be given by a simple explicit formula. This allows us to study flatness of linear pencils on three-dimensional Lie algebras, in particular those related to the argument translation method. Many of them appear to be non-flat.

Original language | English (US) |
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Pages (from-to) | 557-567 |

Number of pages | 11 |

Journal | Differential Geometry and its Application |

Volume | 31 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2013 |

Externally published | Yes |

### ASJC Scopus subject areas

- Analysis
- Geometry and Topology
- Computational Theory and Mathematics