Classification of isoparametric hypersurfaces

等参超曲面的分类

• 批准号: 5407075
• 负责人: Professor Dr. Linus Kramer
• 金额: --
• 依托单位: Department of Mathematics
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2003
• 资助国家: 德国
• 起止时间: 2002-12-31 至 2009-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5407075?language=en
• 关键词: Classification; isoparametric; hypersurfaces

Isoparametric hypersurfaces are hypersurfaces in spheres with constant principal curvatures. The known examples either come from isotropy representations of symmetric spaces or from representations of Clifford algebras. The isoparametric hypersurfaces arising from symmetric spaces are the principal orbits of the isotropy representation of the symmetric space. In contrast to this, most of the examples arising from Clifford algebras do not admit a transitive isometry group. The classification of isoparametric hypersurfaces is one of the major open problems in submanifold geometry. One aim of the project is the classification of isoparametric hypersurfaces with four distinct principal curvatures. Closely connected to isoparametric submanifolds are manifolds which have the same rational cohomology as a projective space: The diffeomorphism classification of such manifolds is an open problem; in particular, the classification of manifolds which are like projective planes is open. We plan to achieve such a classification.

等参超曲面是具有恒定主曲率的球面上的超曲面。已知的例子要么来自对称空间的各向同性表示，要么来自Clifford代数的表示。由对称空间产生的等参超曲面是对称空间各向同性表示的主轨道。与此相反，大多数由Clifford代数产生的例子不承认传递等距群。等参超曲面的分类是子流形几何中主要的开放问题之一。该项目的一个目标是分类具有四个不同主曲率的等参超曲面。与等参子流形紧密相连的是与射影空间具有相同有理上同调的流形，这类流形的微分同态分类是一个开放问题；特别地，像投影平面的流形的分类是开放的。我们计划实现这样的分类。