Theory of Differential Forms and Twistor Structures

微分形式和扭转结构理论

• 批准号: 08454016
• 负责人: SATO Hajime
• 金额: $ 2.94万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454016/
• 关键词: differential form; involutive system; twistor; third order ODE; Grassman structure; ツイスター構造

In this reserch during two years, we have studied the twistor diagram constructed by the head investigator. The diagram gives relations between various geometric structures. We use the theory of involutive system of differential forms and clarified the relations of the diagram with the theory of particles and the field theory. Main purpose of our studies is application of the twistor theory to the classical and quantum mechanics. Especially, we investigated the following geometric strustures in detail : The structure defined by third order ordinary differential equations, projective structures, Grassmann structures, Lie contact structures, etc. We showed that in many cases one equation of a structure is transformed to a simple equation of other geometric structure.As concrete results, the head investigator with Mrs.A.Y.Yoshikawa studied the equivalence problem under contact diffeomorphism of third order ordinary differential equations, proved that the complete invariants are given by two geometric curvatures and decided the concrete form of the curvatures. By the twistor theory, this result is connected to the geometry of the relativity and gives the fundation of projective contact geometry.Further the head investigator studied the Grassmann structures related to the geometry of the system of second order differential equations and made clear the relation between the half flatness and the twistor theory of the null bundles.

在两年的研究中，我们研究了首席研究员构建的扭量图。该图给出了各种几何结构之间的关系。我们运用微分形式对合系统理论，阐明了图与粒子论、场论的关系。我们研究的主要目的是将扭量理论应用于经典力学和量子力学。特别是，我们详细研究了以下几何结构：由三阶常微分方程、射影结构、格拉斯曼结构、李接触结构等定义的结构。我们表明，在许多情况下，结构的一个方程可以转化为一个简单的方程作为具体成果，首席研究员与Mrs.A.Y.Yoshikawa研究了三阶常微分方程接触微分同胚下的等价问题，证明了完全不变量由两个几何给出曲率并确定了曲率的具体形式。通过扭量理论，这一结果与相对论几何联系起来，为射影接触几何奠定了基础。进一步研究了与二阶微分方程组几何相关的格拉斯曼结构，明确了零束的半平坦度和扭转理论。

项目成果

佐藤 肇: "位相幾何(現代数学の基礎2)" 岩波書店, 136 (1996)

佐藤肇：《拓扑学（现代数学基础2）》岩波书店，136（1996）

H.Izeki: "A deformation of flat conformal structures" Trans.Amer.Math.Soc.348. 4939-4964 (1996)

H.Izeki：“平面共形结构的变形”Trans.Amer.Math.Soc.348。

N.Ejiri: "Degenerate minimal surface of finite total curvature in R^N" Kobe J.Math.14. 11-22 (1997)

N.Ejiri：“R^N 中有限总曲率的简并最小曲面”Kobe J.Math.14。

H.Izeki: "A deformation of flat conformal strucutures" Trans.Amer.Math.Soc.348. 4939-4964 (1996)

H.Izeki：“平面共形结构的变形”Trans.Amer.Math.Soc.348。

向井茂: "Brill-Noether理論の非可換化と3次元Fano多様体" 数学. 49. 1-24 (1997)

Shigeru Mukai：“Brill-Noether 理论和三维 Fano 流形的非交换化” 数学 49. 1-24 (1997)