Mathematical Sciences: RUI: Dupin Submanifolds

9504535欧氏n维空间中的超曲面，如果它的主曲率不变，则称它是等参超曲面；可以证明这样的超曲面一定是平面、球面或球面柱面。然而，单位n球面中的等参超曲面是一个有趣的例子的丰富来源；它的完整分类仍然是缺乏的。Dupin超曲面是等参超曲面的推广，它假定主曲率仅沿某些方向为常数。建议的研究将尝试对具有4到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，以及3个不同的主曲率的情况最近由提出者和G.Jensen完成。等参超曲面和Dupin超曲面构成了经典微分几何中一个特别美丽的子域；这些曲面同时是高度对称且非平凡的。

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.