Global Research of Geometry related with Poisson and Contact Manifolds.

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In the first year of the term of the project, we investigated Nambu-Jacobi manifolds and gave a characterization of such manifolds interms of multi-vector fields. This result is written in the preprint Foliations assocaited with Nambu-Jacobi structures which is a joint paper with K. Mikami(Akita University).In the second and the third year of the project, we were concerned with two topics. The one is the Leibniz algebra associated with a Nambu-Poisson manifold. We first observed that given a decomposable integrable p-form, the space of p+1-vector fields on the manifold have a structure of Leibniz algebra. Further we observed that this algebra structure depends only on the diffeomorphism class of the foliation defined by the p-form. Also, there is a natural Leibniz homomorphism from this algebra to the Lie algebra which is formed by the vector fields tangent to the foliation. As in the case of Lie algebras, this extension of algebra corresponds to a 2-dimensional cocycle of a Leibniz cohomology. These results are contained in the paper Y. Hagiwara-Tmizutani "Leibniz algebras associated with foliations" The other is study of the Pfaff system regarding it as a submanifold of the symplectic manifold T^*M. A. typical result of this direction is that the Pfaff system is completely integrable if it is a coisotropic submanifold of T^*M. From this vie point we described the Godbillon-Vey class as a intersection of certain naturally defined multi-vector fields.