Document Type
Article
Publication Date
9-25-2025
Department
Mathematics
Abstract
A hypersurface M in the unit sphere Sn ⊂ Rn+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 Sn. One construction was published in 1989 by Pinkall and Thorbergsson [35], and the second was published in 1989 by Miyaoka and Ozawa [26]. Both types of examples have the property that they do not have constant Lie curvatures (Lie invariants discovered by Miyaoka [24]), 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 Sn. So they are counterexamples to a conjecture of Cecil and Ryan [13, p. 184] in 1985 that every compact proper Dupin hypersurface in Sn is equivalent to an isoparametric hypersurface by a Lie sphere transformation.
DOI
https://doi.org/10.48550/arXiv.2509.21235
Repository Citation
Cecil, Thomas E., "Constructions of Compact Dupin Hypersurfaces with Non-constant Lie Curvatures" (2025). Mathematics and Computer Science Department Faculty Scholarship. 24.
https://crossworks.holycross.edu/math_fac_scholarship/24
Comments
The Mathematics Subject Classification classification numbers are: 53A07, 53A40, 53B25, 53C40, 53C42