Isoparametric and Dupin Hypersurfaces

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2008
• Main Author: Cecil, T.E.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2008
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/149015
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Isoparametric and Dupin Hypersurfaces / T.E. Cecil // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 171 назв. — англ.

A hypersurface Mn−1 in a real space-form Rn, Sn or Hn is isoparametric if it has constant principal curvatures. For Rn and Hn, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Élie Cartan showed in a series of four papers in 1938–1940, the subject is much deeper and more complex for hypersurfaces in the sphere Sn. A hypersurface Mn−1 in a real space-form is proper Dupin if the number g of distinct principal curvatures is constant on Mn−1, and each principal curvature function is constant along each leaf of its corresponding principal foliation. This is an important generalization of the isoparametric property that has its roots in nineteenth century differential geometry and has been studied effectively in the context of Lie sphere geometry. This paper is a survey of the known results in these fields with emphasis on results that have been obtained in more recent years and discussion of important open problems in the field.