Mathematical Sciences: Applications of Lie Sphere Geometry to the Study of Taut and Dupin Submanifolds

数学科学：李球几何在张紧子流形和杜宾子流形研究中的应用

• 批准号: 8706015
• 负责人: Thomas Cecil
• 金额: --
• 依托单位: College of the Holy Cross
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1989-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8706015&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Applications; Lie; Sphere

Thomas Cecil will continue his long term investigations of taut hypersurfaces. The focus of attention will be on finding interrelationships between these, Dupin hypersurfaces and isoparametric hypersurfaces. This research will follow on from recent work, with collaborators, using the techniques of Lie sphere geometry. A taut hypersurface is one on which every Euclidean distance function has the minimum number of critical points required by the Morse inequalities. Such hypersurfaces are tight in the sense that they have minimum absolute total curvature. Tautness is also related to the notion of Dupin hypersurfaces. One of the principal questions to be studied is whether a Dupin hypersurface embedded in the sphere must be taut.

托马斯·塞西尔将继续他对紧绷超曲面的长期研究。关注的焦点将是寻找这些，杜邦超曲面和等参超曲面之间的相互关系。这项研究将在最近的工作的基础上，与合作者一起使用李球几何技术。紧绷超曲面是每一个欧几里得距离函数都具有莫尔斯不等式所要求的最小临界点数目的曲面。这样的超曲面是紧的，因为它们有最小的绝对总曲率。紧度也与杜宾超曲面的概念有关。要研究的主要问题之一是嵌入球中的杜宾超曲面是否必须被拉紧。