Applications of Lie Sphere Geometry to Submanifold Theory

摘要奖：DMS-0071390主要研究者：托马斯E.塞西尔首席研究员和他的合作者，郭信志和加里詹森，将研究子流形的欧几里德空间和领域的背景下，李球几何。 特别感兴趣的是具有特殊曲率性质的子流形。 这些包括等参超曲面，其中有恒定的主曲率，和Dupin超曲面，其中有属性，每个主曲率是恒定的每个相应的曲率曲面。 主要研究的问题是球面上具有四个主曲率的等参超曲面的分类和具有四个或六个主曲率的局部不可约Dupin超曲面的分类。 本研究主要是局部性质的，采用李球几何中的移动标架方法，重点研究一类重要的曲面Dupin曲面，它具有非常特殊的曲率性质。 杜宾曲面的例子有平面、球面、圆柱体和杜宾环，这些曲面近年来在先进的计算机辅助设计中很有用。 Dupin曲面具有高维的类似物，它最早是由法国数学家Elie Cartan在20世纪30年代研究的，在过去的30年里，许多数学家对它进行了广泛的研究。 本文的目标是对这些高维Dupin曲面进行分类。 该提案的另一个重要方面是主要研究者对本科生的指导工作。 在资助期间，三名本科生将获得资助，在与主要研究者自己的研究相关的领域进行夏季指导性独立研究。每个学生都会根据这项研究写一篇荣誉论文。 在过去的十年里，在主要研究者的指导下撰写荣誉论文的学生中，大多数都在攻读数学研究生。 通过这种方式，首席研究员以前的赠款在一些下一代科学家的教育中发挥了重要作用。

AbstractAward: DMS-0071390Principal Investigator: Thomas E. CecilThe principal investigator and his collaborators, Quo-Shin Chiand Gary Jensen, will study submanifolds of Euclidean space andthe sphere within the context of Lie sphere geometry. Ofparticular interest are submanifolds with special curvatureproperties. These include isoparametric hypersurfaces, whichhave constant principal curvatures, and Dupin hypersurfaces,which have the property that each principal curvature is constantalong each of its corresponding curvature surfaces. The mainproblems to be studied are the classification of isoparametrichypersurfaces of the sphere with four principal curvatures, andthe classification of locally irreducible Dupin hypersurfaceswith four or six principal curvatures. This research isprimarily local in nature, using the method of moving frames inLie sphere geometry.This project focuses on an important class of surfaces, Dupinsurfaces, which have very special curvature properties. Examplesof Dupin surfaces are planes, spheres, circular cylinders and thecyclides of Dupin, which have been useful in recent years inadvanced computer-aided design. Dupin surfaces have higherdimensional analogues which were first studied by the greatFrench mathematician Elie Cartan in the 1930's and which havebeen researched extensively by many mathematicians over the pastthirty years. The goal of the proposed research is to classifythese higher dimensional Dupin surfaces. Another importantaspect of the proposal is the principal investigator's mentoringwork with undergraduate students. Over the period of the grant,three undergraduate students will be supported by the grant for asummer of directed independent study in an area related to theprincipal investigator's own research. Each of these studentswill then write an honors thesis based on this study. In thepast ten years, most of the students who have written honorstheses under the principal investigator's supervision have thenpursued graduate study in mathematics. In this way, theprincipal investigator's previous grants have played asignificant role in the education of some of the next generationof scientists.