Global bi-Hamiltonian structure of three dimensional dynamical systems

In this thesis the conditions for the local existence of the bi-Hamiltonian structure corresponding to a non-vanishing vector field on three dimensional manifolds are obtained by using the given vector field alone. It is shown that any non-vanishing vector field on a three dimensional manifold is locally bi-Hamiltonian. Then by working on the equations related with the local existence, the obstructions to extension of local bi-Hamiltonian structure are obtained. After that, these obstructions are expressed in terms of the characteristic classes related with the normal bundle of the given vector field. It is shown that any non-vanishing vector field on a three dimensional manifold is globally bi-Hamiltonian if and only if the Chern class of the normal bundle of the vector field and Bott class of the transversally holomorphic complex codimension one foliation defined by the vector field vanishes.