论文标题
dupin submanifolds in Lie Sphere几何形状（更新版本）
Dupin Submanifolds in Lie Sphere Geometry (updated version)
论文作者
论文摘要
如果$ m^{n-1} $在$ m^{n-1} $上恒定，则欧几里得空间中的HyperSurface $ m^{n-1} $是适当的dupin，并且每个主曲率函数沿其主要叶子的每个叶子都持续不变。本文最初发表于1989年（请参见下面的评论），并在使用移动框架的Lie Sphere几何形状的背景下开发了一种本地研究适当的Dupin Hypersurfaces的方法。从那时起，该方法就可以有效地获得适当的Dupin Hypersurfaces的多个分类定理。该论文的更新版本包含原始博览会以及T.Cecil在2020年（文本中所示）的一些言论，这些言论描述了自原始版本以来现场的进度，以及该领域中一些重要的剩下的开放问题。
A hypersurface $M^{n-1}$ in Euclidean space $E^n$ is proper Dupin if the number of distinct principal curvatures is constant on $M^{n-1}$, and each principal curvature function is constant along each leaf of its principal foliation. This paper was originally published in 1989 (see Comments below), and it develops a method for the local study of proper Dupin hypersurfaces in the context of Lie sphere geometry using moving frames. This method has been effective in obtaining several classification theorems of proper Dupin hypersurfaces since that time. This updated version of the paper contains the original exposition together with some remarks by T.Cecil made in 2020 (as indicated in the text) that describe progress in the field since the time of the original version, as well as some important remaining open problems in the field.
