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Research on Geometric Structures, Schwarzian Derivatives of Maps and Partial Differential Equations

几何结构、映射的施瓦茨导数和偏微分方程研究

基本信息

批准号：
16540085
负责人：
OZAWA Tetsuya
金额：
$ 1.48万
依托单位：
Meijo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16540085/
关键词：
Contact Structure CR structure Conformal Structure Schwarzian Derivative Partial Differential Equation Normal Connection Heisenberg Frame Buncle Zero Curvature Condition 射影構造

项目摘要

1. The existence and the uniqueness of the so-called Tanaka-Webster connection of a CR structure was reproved by using a Heisenberg frame bundle, in consideration that specific connections are expected to play an important role as the Schwarzian derivatives of transformations in various geometric structures.2. On the underlying contact structure of a CR structure, a necessary and sufficient condition of a Hamiltonian function to generate a contact transformation that preserves the CR structure is established.3. su_2 frame bundle was examined in detail, and shown to be useful in CR geometry, for example, in analyzing the Tanaka-Webster connections.4. It is shown that the deformation parameter of CR structure is explained by complex valued functions, which was used to exhibit the Levi form and the connection coefficients in precise form.5. The notion of Schwarzian derivative for higher dimensional contact transformations was established, which was used to solve an equivalence problem of a certain system of partial differential equations.6. The notion of Schwarzian derivative in CR geometry for strictly contact transformations was established. As the result, the fundamental equation of 3-dimensional CR structure was obtained, and a natural Hermitian structure on the solution space was detected. Finally the equivalence problem of 3 dimensional CR structure was solved.

1.利用Heisenberg框架丛重新证明了CR结构的Tanaka-Webster联络的存在性和唯一性，考虑到特定联络在各种几何结构中作为变换的Schwarzian导数起着重要作用.在CR结构的底层接触结构上，建立了Hamilton函数产生保持CR结构的接触变换的充分必要条件. su_2框架束被详细地检查，并且被证明在CR几何学中是有用的，例如，在分析Tanaka-Webster连接中.研究表明，CR结构的变形参数可以用复变函数来解释，并用复变函数来表示Levi形式和连接系数的精确形式.建立了高维接触变换的Schwarzian导数的概念，并将其用于求解一类偏微分方程组的等价问题.建立了CR几何中严格切触变换的Schwarzian导数的概念。结果得到了三维CR结构的基本方程，并在解空间上发现了一个自然的厄米特结构。最后解决了三维CR结构的等效问题。

项目成果

期刊论文数量（7）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

部分複素幾何学と基本方程式

部分复杂几何和基本方程

DOI：
发表时间：
2007
期刊：
影响因子：
0
作者：
Hajime SATO;Tetsuya OZAWA;Hiroshi SUZUKI;小沢哲也
通讯作者：
小沢哲也

Noncommutative Geometry and Physics 2005 : Proceedings of Int'l Sendai-Beijing Joint Workshop,2005.

非交换几何与物理2005：国际仙台-北京联合研讨会论文集，2005。

DOI：
发表时间：
2007
期刊：
影响因子：
0
作者：
Hajime SATO;Tetsuya OZAWA;Hiroshi SUZUKI;小沢哲也;Tetsuya OZAWA;小沢哲也;Tetsuya OZAWA;Watamura et al.(eds.)
通讯作者：
Watamura et al.(eds.)

ouncomplex structures and the tunaamental equations.

复杂的结构和tunaamental方程。