First Order Partial Differential Equation and WEB Geometry

一阶偏微分方程和WEB几何

• 批准号: 08640077
• 负责人: NAKAI Isao
• 金额: $ 1.6万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 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-08640077/
• 关键词: Web; Foliation; Chern onnection; Integrable system; Complex dynamics; Non solvable Dynamics; First Order PDE; Higher Order ODE; 高階偏微分方程式; Web; 一階微分方程式; ルジャンドル; 接触構造; pseudo group; 留数公式

Differential geometry of Web structure is to study the relation of geometric structure of configurations of foliations and its affine connection. In general the connection is not unique but there are finitely many connections. In this research project, I determined all configurations of codimension one foliations for which all those connections are equal. This result generalizes the classical result due to Poincare, Reidemeister and Mayrhofer. And also I showed that in general the mean of curvature forms of all those affine connections is the curvature form defined by Blaschke in 1930's. This observation motivated to apply Web geometry to certain integrable systems. A holonomic partial differential equation on R' is a n-dimensional variety in its projective cotangent bundle. On the variety the contact form restricts to one form, of which the integrable manifolds are the solutions of the equations. Web geometry applies here to extend Bott connection of the foliation by the solutions to unique affine connection on the variety. One of my results tells the mean of the resulting affine connection projected to the base space R' gives Blaschke curvature form. For a certain moduli space of holonomic PDE with complete integrals I showed it is in one to one correspondence with the space of Blaschke curvature forms. This result was announced in the symposium on Web geometry "Journees sur les Tissus" held at Univ. Paul Sabatier, Toulouse in 1996 December, and the paper on the result is in preparation to publish as a part of the lecture note of the symposium. In another vean, I discussed the local problem of the classification problem of the domains in the complex plane with analytic boundaries by using a method in complex dynamical systems.

Web结构的微分几何是研究叶面构型的几何结构与其仿射联系的关系。通常，连接不是唯一的，但存在有限多个连接。在这个研究项目中，我确定了所有这些连接都相等的余维一叶的所有配置。这一结果推广了Poincare，Reidemister和Mayhofer的经典结果。我还证明了所有这些仿射联络的曲率形式的平均是由Blaschke在1930年的S定义的曲率形式。这一观察结果促使我们将Web几何应用于某些可积系统。R‘上的完整偏微分方程系其射影余切丛中的n维簇。接触形式仅限于一种形式，其中可积流形是方程的解。卷筒纸几何应用于此，通过解决各种独特的仿射连接来扩展叶面的Bott连接。我的一个结果告诉我们投影到基本空间R‘上的仿射联络的平均值给出了Blaschke曲率形式。对于具有完全积分的完整偏微分方程模空间，我证明了它与Blaschke曲率形式空间一一对应。这一结果是在大学举行的网络几何学专题讨论会“组织期刊”上宣布的。1996年12月，保罗·萨巴蒂埃，图卢兹，关于这一结果的论文正在准备作为专题讨论会讲稿的一部分出版。在另一篇文章中，我用复动力系统中的一种方法讨论了具有解析边界的复平面中域的分类问题的局部问题。

项目成果

諏訪 立雄: "InAicec of vector fields and Residues of helomorphic foliatiens" Hermann publisher, 203 (1998)

Tatsuo Suwa：“矢量场的 InAicec 和异形叶状体的残基”赫尔曼出版社，203 (1998)

Kawazumi Nariya: "Homology of hyperelliptic mapping class groups for surfaces" Topology and its appl. 76. 203-216 (1997)

Kawazumi Nariya：“曲面超椭圆映射类群的同调”拓扑及其应用。

中居 功: "The classification of ourvilinear angles in the complex plane and the groups of ± holomorphic diffeomorphisms" Annals Math. Toulouse. (1997)

Isao Nakai：“复平面中的线角分类和±全纯微分同胚”年鉴图卢兹（Annals Math）。

泉屋 周一: "Formations of singularities for viscosity solutions of Hamilton-Jacobi eguations" Banach Center Publications. 33. 127-148 (1996)

Shuichi Izumiya：“Hamilton-Jacobi 方程粘度解的奇异性的形成”Banach Center Publications 33. 127-148 (1996)。

Sato Hajime: "Third order ordinary differential equations and Legendre connections" J.Math.Soc.Japan. 50(in press).

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