2020/10/13 by Thomas E. Cecil, Cecil, Thomas E., Shiing-Shen Chern +1
Mathematics · Physics and Astronomy · #53A07 #53A40 #53B25 (Primary) #53C40 #53C42 (Secondary) #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2010.06429
openalex publication_date 2020/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A hypersurface Mn-1 in Euclidean space En is proper Dupin if the number of distinct principal curvatures is constant on Mn-1, and each principal curvature function is constant along each leaf of its principal foliation. This paper was originally published in 1989 (see Comments below), and it develops a method for the local study of proper Dupin hypersurfaces in the context of Lie sphere geometry using moving frames. This method has been effective in obtaining several classification theorems of proper Dupin hypersurfaces since that time. This updated version of the paper contains the original exposition together with some remarks by T.Cecil made in 2020 (as indicated in the text) that describe progress in the field since the time of the original version, as well as some important remaining open problems in the field.