Dirac-nijenhuis Structure

This paper generalizes the notion of Poisson-Nijenhuis manifolds (PN manifolds) and presymplectic-Nijenhuis manifolds (N manifolds) and defines the notion of Dirac-Nijenhuis structures on manifolds. Meanwhile we shall discuss the properties of the structures. All the discussions above will extend to the case of Lie bialgebroids.First of all the concept of compatible Nijenhuis tensors and deformed Lie bialgebroids are denned. With this in mind we define the DN structures on manifolds as follows: Suppose that N1 and N2 are compatible Nijenhuis tensors on a manifold P. Lis a. Dirac structure both on the canonical Lie bialgebroid (TP, T*P) and on ((TP)', T*P), the deformed Lie bialgebroid of (TP, T*P) induced by N = N1N2. So is L1 = (N1,N2)(L). Then we call (L,N1,N2) a Dirac-Nijenhuis structure (DN structure) on P. The manifold P endowed with a DN structure (L, N1 , N2) is called a Dirac-Nijenhuis manifold (DN manifold) and is denoted by (P,L,N1,N2). One of the important conclusions in this paper is the two necessary and sufficient conditions for a structure to be a DN structure in terms of the characteristic pairs and the dual characteristic pairs of Dirac structures.Several examples of DN manifolds are presented and we shall discuss the properties of some examples from which we derive the conclusion that both PN manifolds and N manifolds are the special cases of DN manifolds.To study the submanifolds of DN manifolds we shall discuss the submanifolds of Dirac manifolds on the base of which we shall prove that under certain conditions the DN structures on manifolds can induce naturally DN structures on the submanifolds and that deforming Dirac structures and inducing new Dirac structures on the submanifolds by the original Dirac structures are commutative.The reduction theorem of DN manifolds is obtained by using that of Dirac manifolds.Against a background of the symplectic vector fields on symplectic manifolds, the Poisson vector fields on Poisson manifolds and the fundamental vector fields on PN manifolds, we introduce the notion of the fundamental vector fields on DN manifolds and study its properties. Leaving the geometric structure invariant is a common characteristic of the first three kinds of vector fields. As an analogue of the three vector fields the fundamental vector fields of DN manifolds preserve the DN structures. However, due to the complexity of DN structures, preserving the structures is not simple as preserving the symplectic structures, Poisson structures and PN structures. As examples we obtain all the fundamental vector fields for the Poisson case of DN manifolds and the presymplectic case of DN manifolds, on the base of which we reach an important conclusion that the spaces of fundamental vector fields of PN manifolds coincide with that of PN manifolds which are regarded as DN manifolds. We shall also discuss the relation between the fundamental vector fields on DN manifolds and that on submanifolds as well as the relation between the fundamental vector fields on DN manifolds and that on the reductive DN manifolds.At the end of chapter 3 we are going to define the DN mappings between two DN manifolds following which some examples of DN mappings are given. DN mappings, the smooth mappings preserving DN structures, are analogues of symplectic mappings, Poisson mappings and Dirac mappings.Chapter 4 is the continuation of the work in the previous chapter. Since Dirac structures on manifolds have been generalized to the case of Lie bialgebroids, we do the same for DN structures on manifolds. Meanwhile we shall discuss the properties of DN structures on Lie bialgebroids. Owing to the fact that for a general Lie bialgebroid (A, A*) the Lie algebroid structure in A* is nontrivial, DN structures on Lie bialgebroids are more complicated than that on manifolds. We have obtained the necessary and sufficient conditions for a structure to be a DN structure on a bialgebroid. Nevertheless because the foundation of this chapter is Lie bialgebroids rather than manifolds, the given conditions are weaker th...