Mathematical Sciences: RUI: Dupin Submanifolds

数学科学：RUI：杜宾子流形

• 批准号: 9504535
• 负责人: Thomas Cecil
• 金额: $ 7.5万
• 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: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504535&HistoricalAwards=false
• 关键词: Mathematical; Sciences; RUI; Dupin; Submanifolds

9504535 Cecil A hypersurface in Euclidean n-space is said to be isoparametric if it has constant principal curvatures; it can be shown that such a hypersurface must be a plane, a sphere or a spherical cylinder. Isoparametric hypersurfaces in the unit n-sphere, however, are a rich source of interesting examples; its complete classification is still lacking. Dupin hypersurfaces generalize isoparametric hypersurfaces in that the principal curvatures are assumed to be constant only along certain directions. The proposed research will attempt to classify Dupin hypersurfaces with 4 or 6 distinct principal curvatures. Cartan showed that such a hypersurface can have 1, 2, 3, 4, or 6 distinct principal curvatures; when there are 1 or 2 distinct principal curvatures the classification problem iq routine, and the case of 3 distinct principal curvatures was recently done by the proposer and G. Jensen. Isoparametric and Dupin hypersurfaces form a particularly beautiful subarea of classical differential geometry; these surfaces are highly symmetric and nontrivial at the same time.

9504535塞西尔 欧氏n-空间中的超曲面称为等参的，如果它具有常数的主曲率;可以证明这样的超曲面必须是平面、球面或球柱。然而，单位n-球面中的等参超曲面是有趣例子的丰富来源;它的完整分类仍然缺乏。Dupin超曲面是等参超曲面的推广，它的主曲率仅在沿着某些方向保持不变.本文将尝试对具有4个或6个不同主曲率的Dupin超曲面进行分类。Cartan证明了这样的超曲面可以有1、2、3、4或6个不同的主曲率;当有1或2个不同的主曲率时，分类问题iq例程，而3个不同的主曲率的情况最近由提议者和G.詹森。 等参超曲面和杜宾超曲面形成了经典微分几何中一个特别美丽的子区域;这些曲面同时具有高度对称性和非平凡性。