Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global Existence of Bi-Hamiltonian Structures on Orientable
  Three-Dimensional Manifolds   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.

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The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
27
The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat