### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 055, 17 pages      arXiv:1612.06996      https://doi.org/10.3842/SIGMA.2017.055

### Global Existence of Bi-Hamiltonian Structures on Orientable Three-Dimensional Manifolds

Melike Işim Efe and Ender Abadoğlu
Yeditepe University, Mathematics Department, İnȯnu Mah. Kayışdağı Cad. 326A, 26 Ağustos Yerleşimi, 34755 Ataşehir İstanbul, Turkey

Received December 21, 2016, in final form July 04, 2017; Published online July 14, 2017

Abstract
In this work, we show that an autonomous dynamical system defined by a nonvanishing vector field on an orientable three-dimensional manifold is globally bi-Hamiltonian if and only if the first Chern class of the normal bundle of the given vector field vanishes. Furthermore, the bi-Hamiltonian structure is globally compatible if and only if the Bott class of the complex codimension one foliation defined by the given vector field vanishes.

Key words: bi-Hamiltonian systems; Chern class; Bott class.

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