Poisson-nijenhuis Manifolds And The Dirac Structure

In this paper,we introduce the definition of Lie bialgebroid and its Dirac structures firstly.Then we particularly discuss the tangent Lie bialgebroid,based on the conclusion and thorems given by A.Weinstein,Z-J.,Liu.In section two,we study the Nijenhuis tensor ,which causes the deformation of Poisson tensor.View Poisson-Nijenhuis manifold as a special case of bihamiltonian manifold,we will show some special anduseful properties of Poisson-Nijenhuis manifold . Meanwhile, we will show the deformation Lie bialgebroid.These are the necessary preparations for the later discussion.With the if and only if condition of the condition when a maximally isotropic subbundle is a Dirac structure,we particularly discuss some Lie bialgebroids and its Dirac structures in the section three.Moreover,we get the similar conclusions and theorems.From these,we know more properties of Poisson-Nijenhuis manifold.The last,we study the properties of the basic vector field and 1-form in the Poisson-Nijenhuis manifold,making a systematic organization of these contents and giving some appropriate adds.Therefore ,we unified with the conclusion of the Poisson vector field on Poisson manifold in reform . Considered the relationship between the Dirac structures of Poisson-Nijenhuis manifold and the basic vector field , we proof that the basic vector field can keep the Dirac structures of Poisson-Nijenhuis manifold which discussed before.