First Order Partial Differential Equation and WEB Geometry

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。

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.