Constructions of Compact Dupin Hypersurfaces with Non-constant Lie Curvatures

A hypersurface $M$ in the unit sphere $S^n \subset {\bf R}^{n+1}$ is Dupin if along each curvature surface of $M$, the corresponding principal curvature is constant. If the number $g$ of distinct principal curvatures is constant on $M$, then $M$ is called proper Dupin. In this expository paper, we give a detailed description of two important types of constructions of compact proper Dupin hypersurfaces in $S^n$. One construction was published in 1989 by Pinkall and Thorbergsson, and the second was published in 1989 by Miyaoka and Ozawa. Both types of examples have the property that they do not have constant Lie curvatures (Lie invariants discovered by Miyaoka), which are the cross-ratios of the principal curvatures, taken four at a time. Thus, these examples are not equivalent by a Lie sphere transformation to an isoparametric (constant principal curvatures) hypersurface in $S^n$. So they are counterexamples to a conjecture of Cecil and Ryan in 1985 that every compact proper Dupin hypersurface in $S^n$ is equivalent to an isoparametric hypersurface by a Lie sphere transformation.