Bi-Hamiltonian structures on three dimensional manifolds and eigenvectors of curl operator

In 2017 Işim Efe and Abadoğlu showed that a dynamical system defined by a non-vanishing 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. Further- more, it is possible that 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 van- ishes. In this thesis, a dynamical system on an orientable three dimensional manifold which is globally bi-Hamiltonian which are not globally compatible is investigated. For this, S3, which is a good example of three-dimensional manifolds, has been chosen also the cohomol- ogy groups of S3 provide the desired characteristic classes. In order to find suitable Poisson structures on S3, well-known Maurer-Cartan forms are defined. It has been shown that the eigenforms of the curl operator provide the bi-Hamiltonian structures globally, but they are not globally compatible.