Geodesic
Global Existence of Bi-Hamiltonian Structures on Orientable Three-Dimensional Manifolds
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: bi-Hamiltonian systems; Chern class; Bott class. 
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     title = {Global {Existence} of {Bi-Hamiltonian} {Structures} on {Orientable} {Three-Dimensional} {Manifolds}},
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Melike Išim Efe; Ender Abadoğlu. Global Existence of Bi-Hamiltonian Structures on Orientable Three-Dimensional Manifolds. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a54/

[1] Asuke T., “A remark on the Bott class”, Ann. Fac. Sci. Toulouse Math., 10 (2001), 5–21 | DOI | MR | Zbl

[2] Bott R., “Lectures on characteristic classes and foliations”, Lectures on Algebraic and Differential Topology, Second Latin American School in Math. (Mexico City, 1971), Lecture Notes in Math., 279, Springer, Berlin, 1972, 1–94 | DOI | MR

[3] Dibag I., “Decomposition in the large of two-forms of constant rank”, Ann. Inst. Fourier (Grenoble), 24 (1974), 317–335 | DOI | MR | Zbl

[4] Gümral H., Nutku Y., “Poisson structure of dynamical systems with three degrees of freedom”, J. Math. Phys., 34 (1993), 5691–5723 | DOI | MR | Zbl

[5] Haas F., Goedert J., “On the generalized Hamiltonian structure of $3$D dynamical systems”, Phys. Lett. A, 199 (1995), 173–179, arXiv: math-ph/0211035 | DOI | MR | Zbl

[6] Hernández-Bermejo B., “New solutions of the Jacobi equations for three-dimensional Poisson structures”, J. Math. Phys., 42 (2001), 4984–4996 | DOI | MR | Zbl

[7] Ince E. L., Ordinary differential equations, Dover Publications, New York, 1944 | MR | Zbl

[8] Laurent-Gengoux C., Pichereau A., Vanhaecke P., Poisson structures, Grundlehren der Mathematischen Wissenschaften, 347, Springer, Heidelberg, 2013 | DOI | MR

[9] Magri F., “A simple model of the integrable Hamiltonian equation”, J. Math. Phys., 19 (1978), 1156–1162 | DOI | MR | Zbl

[10] Olver P. J., “Canonical forms and integrability of bi-Hamiltonian systems”, Phys. Lett. A, 148 (1990), 177–187 | DOI | MR

[11] Santoprete M., “On the relationship between two notions of compatibility for bi-Hamiltonian systems”, SIGMA, 11 (2015), 089, 11 pp., arXiv: 1506.08675 | DOI | MR | Zbl

[12] Stiefel E., “Richtungsfelder und Fernparallelismus in $n$-dimensionalen Mannigfaltigkeiten”, Comment. Math. Helv., 8 (1935), 305–353 | DOI | MR

[13] Weinstein A., “The local structure of Poisson manifolds”, J. Differential Geom., 18 (1983), 523–557 | DOI | MR | Zbl