A hypersurface M in Rn 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 Rn or Sn . The theory of compact proper Dupin hypersurfaces in Sn is closely related to the theory of isoparametric hypersurfaces in Sn, and many important results in this field concern relations between these two classes of hypersurfaces. This problem was formulated in 1985 in a conjecture of Cecil and Ryan [17, p. 184], which states that every compact, connected proper Dupin hypersurface M ⊂ Sn is equivalent to an isoparametric hypersurface in Sn by a Lie sphere transformation. This paper gives a survey of progress on this conjecture and related developments.
Cecil, Thomas E., "Compact Dupin Hypersurfaces" (2021). Mathematics Department Faculty Scholarship. 12.