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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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