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.

在两年内，我们研究了由负责人调查员构建的扭曲图图。该图给出了各种几何结构之间的关系。我们使用差异形式的涉及系统的理论，并阐明了该图与粒子理论和田间理论的关系。我们研究的主要目的是将扭曲理论应用于经典和量子力学。特别是，我们详细研究了以下几何强度：由三阶的普通微分方程，投影结构，格拉曼恩结构，谎言接触结构等定义的结构。我们表明，在许多情况下，结构的方程式转换为其他几何结构的简单方程。方程式证明了完整的不变性由两个几何曲率给出，并决定了曲率的混凝土形式。通过Twistor理论，该结果与相对性的几何形状有关，并给出了投射接触几何形状的资金。负责人研究员研究了与二阶微分方程系统的几何形状相关的Grassmann结构，并清楚地清楚了一半平面与无效的曲折理论之间的关系。

项目成果

H.Sato: "Third order ordinar differential equation and Legendre connections" J.Math.Soc.Japan. to appear (1998)

H.Sato：“三阶常微分方程和勒让德联系”J.Math.Soc.Japan。

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 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。

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.Sato, A.Y.Yoshikawa: "Third order ordinary differential equation and Legendre connections" J.Math.Soc.Japan. (to appear). (1998)

H.Sato，A.Y.Yoshikawa：“三阶常微分方程和勒让德联系”J.Math.Soc.Japan。