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Global Research of Geometry related with Poisson and Contact Manifolds.

与泊松和接触流形相关的几何的全球研究。

基本信息

批准号：
11640060
负责人：
MIZUTANI Tadayoshi
金额：
$ 2.3万
依托单位：
Saitama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

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.

在项目的第一年，我们研究了Nambu-Jacobi流形，并给出了这种流形在多向量场项下的表征。这一结果写在与K. Mikami（秋田大学）合著的预印本《Foliations associated with Nambu-Jacobi structures》中。在项目的第二年和第三年，我们关注了两个主题。一个是与南布泊松流形相关的莱布尼茨代数。我们首先观察到给定一个可分解可积的p型，流形上的p+1向量场空间具有莱布尼茨代数结构。进一步我们观察到这个代数结构只依赖于由p型定义的叶理的微分同构类。而且，从这个代数到李代数有一个自然的莱布尼茨同态，它是由与叶理相切的向量场形成的。在李代数的情况下，这个代数的扩展对应于一个莱布尼茨上同调的二维环。这些结果载于Y. Hagiwara-Tmizutani的论文“与叶相关的莱布尼兹代数”。另一个是将Pfaff系统作为辛流形T^*M的子流形的研究。a .这个方向的典型结果是，如果Pfaff系统是T^*M的各向同性子流形，则它是完全可积的。从这个观点出发，我们将哥德亿维类描述为某些自然定义的多向量场的交集。