Compact Dupin Hypersurfaces

A hypersurface $M$ in ${\bf R}^n$ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if the number of distinct principal curvatures is constant on $M$, i.e., each continuous principal curvature function has constant multiplicity on $M$. These conditions are preserved by stereographic projection, so this theory is essentially the same for hypersurfaces in ${\bf R}^n$ or $S^n$. The theory of compact proper Dupin hypersurfaces in $S^n$ is closely related to the theory of isoparametric hypersurfaces in $S^n$, and many important results in this field concern relations between these two classes of hypersurfaces. In 1985, Cecil and Ryan conjectured on p. 184 of the book, "Tight and Taut Immersions of Manifolds," that every compact, connected proper Dupin hypersurface $M \subset S^n$ is equivalent to an isoparametric hypersurface in $S^n$ by a Lie sphere transformation. This paper gives a survey of progress on this conjecture and related developments.