Research on Poisson manifolds and related structures on manifolds.

在该项目的第二年，我们继续研究2-矢量pi定义的平面场的性质。这种平面场的一个典型例子是与泊松结构相关的切辛叶理。在调查过程中，我们获得了以下新的见解。(1)在第一年，我们证明了;当dim M = 2kappa + 1，如果[pi，pi^<kappa>] <double plus>0，pi定义了一个接触平面场。这一次我们证明了M的一个联络的存在，使得Divpi（w.r.t.该连接）是接触结构的（Reel）向量场。(2)若pi定义一个（正则）Poisson结构，则Divpi是Poisson 1-上圈，并且是伴随叶理的模类（在通常的特征类记法中= h_1）的象，这一结果表明在某些情况下可以用Poisson结构定义奇异叶理的特征类.作为第一次尝试在这个方向上，我们拿起左变异泊松结构的李群和尝试。描述了三维叶invarinat（h_3）。以.不同方向的奇异物体，我们研究了流形上的Dirac结构。我们还需要一点时间来整合这些研究，并获得有关的结果。

In the second year of the term of the project, we continued to investigate the prop- erties of the plane field which a 2-vector pi defines. A typical example of such plane field is the tangentially symplectic foliation associated with a Poisson structure. In the course of investigation, we obtained the following new insight. (1) In the first year, we showed ; when dim M = 2kappa + 1, if [pi, pi^<kappa>]<double plus> 0, pi defines a contact plane field. This time we have proved the existence of a connection of M such that Divpi (w.r.t. this connection) is a Reel) vector field of the contact structure. (2) If pi defines a (regular) Poisson sturcture, Divpi is a Poisson 1-cocycle and is the image of a modular class (= h_1 in the usual notation of characteristic classes) of the associated folation.This result suggest the possibility of the definition of characteristic classes of singular foliations in terms of Poisson structures in some cases. As a first attempt in this direction, we picked up the left in variant Poisson structures of Lie groups and tried. to descripe the 3-dimensional leaf invarinat (h_3). In a. different direction of singular objects, we studied Dirac structure on manifoIds. A little more time is needed for us to unite these studies arid obtain adenite results.