Isoparametric and Dupin Hypersurfaces

A hypersurface $M^{n-1}$ in a real space-form ${\bf R}^n$, $S^n$ or $H^n$ is isoparametric if it has constant principal curvatures. For ${\bf R}^n$ and $H^n$, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Elie Cartan showed in a series of four papers in 1938-1940, the subject is much deeper and more complex for hypersurfaces in the sphere $S^n$. A hypersurface $M^{n-1}$ in a real space-form is proper Dupin if the number $g$ of distinct principal curvatures is constant on $M^{n-1}$, 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.