论文标题
使用Lie Sphere几何形状在$ {\ bf r}^n $中研究Dupin Hypersurfaces
Using Lie Sphere Geometry to Study Dupin Hypersurfaces in ${\bf R}^n$
论文作者
论文摘要
$ {\ bf r}^n $或$ s^n $中的高表情$ m $，如果沿每个曲率表面，相应的主曲率是恒定的，则据说为dupin。如果每个主要曲率在$ m $上具有恒定的多重性，即，不同主曲线的数量在$ m $上是恒定的，则据说Dupin Hypersurface是适当的Dupin。 Dupin和适当的Dupin Hypersurfaces在$ {\ bf r}^n $或$ s^n $中的概念可以推广到Lie Sphere几何形状的设置，并且在Lie Sphere Transformations中很容易看出这些属性是不变的。这使得Lie Sphere几何形状成为研究Dupin Hypersurfaces的有效环境，并且已经获得了适当的Dupin Hypersurfaces的许多分类以进行谎言球体的转化。在这些注释中，我们详细介绍了此方法，以研究$ {\ bf r}^n $或$ s^n $中的Dupin Hypersurfaces，包括几个基本结果的证明。
A hypersurface $M$ in ${\bf R}^n$ or $S^n$ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on $M$, i.e., the number of distinct principal curvatures is constant on $M$. The notions of Dupin and proper Dupin hypersurfaces in ${\bf R}^n$ or $S^n$ can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective setting for the study of Dupin hypersurfaces, and many classifications of proper Dupin hypersurfaces have been obtained up to Lie sphere transformations. In these notes, we give a detailed introduction to this method for studying Dupin hypersurfaces in ${\bf R}^n$ or $S^n$, including proofs of several fundamental results.
