Geometry of Partial Differential Equation, Spin Structure and Twistor Theory

偏微分方程几何、自旋结构和扭量理论

• 批准号: 12304004
• 负责人: SATO Hajime
• 金额: $ 17.09万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12304004/
• 关键词: PDE; Twistor thory; Grassmannian structure; Projective contact; Lagrangean; 3^<rd> order ODE; グラスマン構造; カルタン接続; 3階の常微分方程式; 接触幾何学; 球面幾何学; 微分方程式の幾何; シュワルツ微分

As concrete examples of geometric structures related to system of geometric equations, we investigated the following pair structures by using the twistor correspondence.a) projective and Grassmannian b) projective contact and Lagrangean c) Lie contact and Lorentzian d) pure spinor and neutral structuresWe studied in detail the structures, got invariants and classified them. Further we extended the investigations to many different directions.As for a), the head investigator and Machida worked together to obtain complete invariants as elements of Spencer cohomology.Concerning b), the joint work of the head and Yoshikawa was the starting point of the investigation. Ozawa and the head found a system of partial differential equations, which gives a concrete contact transformation.As for c) and d), we got the twistor diagrams and the relation of invariants. As a new extension of the study of differential equations, we got a fundamental system for the conformal structure, which may have many applications in geometry.

作为与几何方程组相关的几何结构的具体例子，我们使用扭量对应研究了以下对结构。a）射影和格拉斯曼 b）射影接触和拉格朗日 c）李接触和洛伦兹 d）纯旋量和中性结构我们研究了详细了解结构，获得不变量并对它们进行分类。进一步，我们将研究扩展到许多不同的方向。对于a），首席研究员和町田共同努力获得了作为斯宾塞上同调元素的完全不变量。关于b），首席研究员和吉川的共同工作是研究的起点。调查。小泽和负责人找到了一个偏微分方程组，给出了具体的接触变换。对于c)和d)，我们得到了扭量图和不变量关系。作为微分方程研究的新延伸，我们得到了共形结构的基本系统，它在几何中可能有许多应用。

项目成果

Hiroshi Ohta: "Obstruction to and deformation of Lagrangian intersection Floer cohomology"Proceeding of KIAS workshop on "Symplectic Geometry and Mirror Symmetry". 281-309 (2001)

Hiroshi Ohta：“拉格朗日交集Floer上同调的阻碍和变形”KIAS“辛几何与镜像对称”研讨会论文集。

Takeshi Sasaki: "The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces II"Journal of Physics A : Mathematical and General. 34. 2319-2328 (2001)

Takeshi Sasaki：“标记立方曲面模空间上复双曲结构的均匀化微分方程 II”物理学杂志 A：数学与一般。

Yoshinori Machida, Hajime Sato: "Twistor thory of manifolds with Grassmannian structures"Nagoya Math.J.. 160. 17-102 (2000)

Yoshinori Machida、Hajime Sato：“具有格拉斯曼结构的流形的扭转理论”Nagoya Math.J.. 160. 17-102 (2000)

Hajime Sato: "Integrability of Contact Schwarzian Derivatives and its Linearization"Proceeding of "Geometry, Integrability and Quantization. 1. 225-228 (2000)

Hajime Sato：“接触 Schwarzian 导数的可积性及其线性化”《几何、可积性和量化》论文集。1. 225-228 (2000)

Tetsuya Ozawa, Hajime Sato: "Contact transformations and their Schwarzian derivatives"Advanced Studies in Pure Math.. 37. 337-366 (2002)

Tetsuya Ozawa、Hajime Sato：“接触变换及其 Schwarzian 导数”纯数学高级研究.. 37. 337-366 (2002)