Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 062, 28 pages      arXiv:0809.1433      https://doi.org/10.3842/SIGMA.2008.062
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Isoparametric and Dupin Hypersurfaces

Thomas E. Cecil
Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610, USA

Received June 24, 2008, in final form August 28, 2008; Published online September 08, 2008

Abstract
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.

Key words: isoparametric hypersurface; Dupin hypersurface.

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