Mathematical Sciences: RUI: Lie Sphere Geometry and Dupin Submanifolds

数学科学：RUI：李球几何和杜宾子流形

• 批准号: 8907366
• 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: Continuing grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1991-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8907366&HistoricalAwards=false
• 关键词: Mathematical; Sciences; RUI; Lie; Sphere

The Dupin condition can be naturally formulated in Lie sphere geometry. It is known to be invariant under certain Lie transformations. Using this knowledge several theorems were proven which concern hypersurfaces whose principal curvatures are constant along their corresponding curvature surfaces. Recently the principal investigator and a collaborator successfully used these transformations to partially solve the problem of classifying all Dupin submanifolds. The principal investigator will continue his work on this classification. Spheres and tori as they appear in three-dimensional space are examples of what mathematicians call two-dimensional submanifolds. These are Dupin if they satisfy certain curvature requirements. The principal investigator has made substantial progress towards the classification of such submanifolds. He will continue his work on this deep and renowned problem.

在李球几何中，Dupin条件可以很自然地表述出来。已知它在某些李变换下是不变的。利用这一知识，证明了几个关于主曲率沿其相应曲率面为常数的超曲面的定理。最近，首席研究员和合作者成功地利用这些变换部分解决了所有Dupin子流形的分类问题。首席研究员将继续他的分类工作。出现在三维空间中的球体和环面是数学家所说的二维子流形的例子。如果它们满足一定的曲率要求，它们就是杜邦。首席研究员在这类子流形的分类方面取得了实质性进展。他将继续研究这个深刻而著名的问题。