Dirac-Nijenhuis结构

【作者】 刘宝康；

【导师】 贺龙光；

【作者基本信息】 首都师范大学 ， 基础数学， 2004， 博士

【摘要】 本文将推广Poisson-Nijenhuis流形（PN流形）和予辛-Nijenhuis流形（ΩN流形）的概念，建立流形上的Dirac-Nijenhuis结构（DN结构）的概念，讨论其性质，并把上述内容推广到李双代数胚上去。 文中首先定义相容的Nijenhuis张量的概念和形变李双代数胚的概念。在此基础上，定义流形上的DN结构如下：设N₁和N₂是流形P上的相容的Nijenhuis张量，L是标准李双代数胚（TP，T^*P）上的Dirac结构，也是由N=N₁N₂诱导的、（TP，T^*P）的形变李双代数胚（（TP）′，T^*P）上的Dirac结构。L₁=（N₁，N₂）（L）亦然。则称（L，N₁，N₂）为流形P上的Dirac-Nijenhuis结构，简称为DN结构。带有DN结构（L，N₁，N₂）的流形P称为Dirac-Nijenhuis流形，简称为DN流形。记为（P，L，N₁，N₂）。利用Dirac结构的特征对和对偶特征对，我们将建立DN结构的两个充要条件。这是本文的重要结论之一。 我们将给出了DN流形的几个例子，并讨论某些例子的性质。由此得出PN流形和QN流形均为DN流形之特例的结论。 为研究DN流形的子流形，我们首先讨论了Dirac流形的子流形。其次，证明了：流形上的DN结构在一定条件下，可以在子流形上自然诱导出一个DN结构，且对Dirac结构的形变和Dirac结构在子流形上诱导出新的Dirac结构这两个手续可换。 在讨论Dirac流形约化的基础上，我们证明了DN流形的约化定理。 我们还特定义DN流形上的基本向量场，研究其主要性质。这是以辛流形上的辛向量场、Poisson流形上的Poisson向量场和PN流形上的基本向量场为背景的。上述三种向量场的共同特征是它们各自保持相应的几何结构。DN流形上的基本向量场做为上述三种向量场的类似物，它保持DN结构，但由于DN结构较为复杂，因而保持DN结构这一特点的表述也相应地比较复杂。做为例子，我们对DN流形的两个例子（Poisson情形的DN流形和予辛情形的DN流形），得出了其上的基本向量场的全体。在此基础上，我们得到如下重要结论：PN漉形的基本向量场空闻和PN流形做为DN流形，其上的基本向量场空间，二者是重合的。另外，我们还对DN流形上的基本向量场和子流形上的基本向量场之同的关系以及DN流形上的基本向量场与约化DN流形上的基本向量场之间的关系进行了讨论。 在第三章的最后，我们将定义DN流形之间的DN映射，并给出了若干例子。DN映射的概念是辛映射、Poisson映射和Dirac映射的类似物，它是保持DN结构的C^∞-映射。 第四章是上一章工作的继续和发展。由于Dirac结构已被推广到了李双代数胚上，因而本文在流形上的DN结构的基础上，把该概念也推广到了李双代数胚上，并讨论了相应的性质。由于在一般李双代数胚（A，A^*）中，A^*的李代数胚结构为非平凡的，因而DN结构从流形发展到李双代数胚上时，具有一定的复杂性。我们得出了李双代数胚上的DN结构的充要条件。然而，由于本章讨论的基础是李双代数胚而非流形，因而已知条件弱于上一章。这导致本章的很多结论比上一章中的相应结论复杂。更重要的是，对于一个李双代数胚（A，A^*）及A上的Nijenhuis张量N，用N形变A所得的形变李代数胚A′与A^*不一定能构成一个李双代数胚。因而在定义李双代数胚（A，A^*）上的DN结构时，需假定（A′，A^*）是李双代数胚。这是在讨论DN流形时不曾遇到的新问题，也是第四章与第三章的一个明显的不同点。因此，我们特别给出了（A′，A^*）成为李双代数胚的两个充要条件。这两个充要条件是分别从A′和A^*之间的相容条件d_*[X，Y]′=[d_*X，Y]′+[X，d_*Y]′（X、Y∈Г（A））和d′[α，β]_*=[d′α，β]_*+[α，d′β]_*（α、β∈Г（A^*））出发得到的。因而这两个充要条件在形式上是对偶的。但其证明的复杂性却相去甚远。原因是，A′为A的形变李代数胚，而A^*却未经形变，二者地位不等。 如同DN流形的情形一样，我们还给出了李双代数胚上的DN结构的一些例子。 最后，我们定义了李双代数胚上的DN结构的基本截面的概念，说明了流形P上的DN结构（视为李双代数胚（TP，T^*P）上的DN结构）的基本截面与DN流形P上的基本向量场的关系，给出了DN波形上的基本向量场的等价定义。另外，和DN流形的情况类似，我们仍能证明基本截面保持DN结构，且基本截面空间是一个李代数.更多还原

【Abstract】 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更多还原