First Order Partial Differential Equation and WEB Geometry

网结构微分几何是研究叶理构形的几何结构与其仿射联系之间的关系。一般来说，连接不是唯一的，而是有许多连接。在这个研究项目中，我确定了余维1叶理的所有配置，所有这些连接都是相等的。这个结果推广了Poincare，Reidemeister和Mayrhofer的经典结果.并且证明了一般情况下所有仿射联络的曲率形式的平均值是Blaschke在1930年定义的曲率形式。这一观察动机应用Web几何某些可积系统。R'上的完整偏微分方程是其射影余切丛中的一个n维簇。在变分上，切触形式限定为一种形式，其中的可积流形是方程的解。本文应用网几何的方法，将叶理的Bott联络推广到簇上唯一仿射联络的解。我的一个结果告诉平均的仿射联络投影到基空间R'给出Blaschke曲率形式。对于具有完全积分的完整偏微分方程的模空间，我证明了它与Blaschke曲率形式空间是一一对应的。这一结果被宣布在专题讨论会上的Web几何形状“alcohenes sur les Tissus”在大学举行的保罗Sabatier，图卢兹在1996年12月，和文件的结果是在准备出版的一部分，讲座笔记的专题讨论会。另一方面，利用复动力系统中的一种方法，讨论了复平面上具有解析边界的区域的分类问题的局部性问题。

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.