2021/01/13 by Thomas E. Cecil, Cecil, Thomas E. · 1 citation
Mathematics · #53B25 #53C40 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53B25 #msc:53C40 #msc:53C42
paper · pdf · doi:10.48550/arxiv.2101.05316
29 pages. arXiv admin note: substantial text overlap with arXiv:2011.11432
arxiv created 2021/04/13 · arxiv updated 2021/10/13
A hypersurface M in \bf 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 \bf 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. 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 ⊂ 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.