Mathematical Sciences: RUI: Lie Sphere Geometry and Dupin Submanifolds

杜宾条件可以在李球几何中自然地表述。已知它在某些李变换下是不变的。利用这一知识，证明了几个定理，其中涉及超曲面的主曲率是恒定的沿着其相应的曲率曲面。最近，主要研究者和合作者成功地使用这些变换来部分解决所有Dupin子流形的分类问题。首席研究员将继续进行这一分类工作。在三维空间中出现的球面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，则它们是杜宾。主要研究者已经取得了实质性的进展，对这种子流形的分类。他将继续就这一深刻而著名的问题开展工作。

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.