First Order Partial Differential Equation and WEB Geometry

卷材结构的微分几何是研究叶状构象的几何结构与其仿射连接的关系。一般来说，连接不是唯一的，但有有限多的连接。在这个研究项目中，我确定了所有这些连接相等的余维叶的所有构型。这个结果推广了庞加莱、赖德梅斯特和梅罗弗的经典结果。我还展示了一般来说所有仿射连接的曲率形式的平均值是由Blaschke在1930年定义的曲率形式。这种观察促使我们将Web几何应用于某些可积系统。在R'上的完整偏微分方程是它的射影余切束的n维变化。在变化上，接触形式限制为一种形式，其中可积流形是方程的解。通过对品种上的唯一仿射连接的求解，本文应用网状几何来扩展叶理的博特连接。我的一个结果告诉我们，由此产生的仿射连接投影到基空间R'的平均值给出了Blaschke曲率形式。对于具有完整积分的完整偏微分方程的模空间，我证明了它与Blaschke曲率形式的空间是一一对应的。这一结果于1996年12月在图卢兹Paul Sabatier大学举行的网络几何研讨会“Journees sur les Tissus”上公布，有关这一结果的论文正准备作为该研讨会的讲稿的一部分发表。另一方面，利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上的域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.