Affiliation:
1. Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610, USA
Abstract
A hypersurface M in Rn or Sn 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 each principal curvature has constant multiplicity on M, i.e., the number of distinct principal curvatures is constant on M. The notions of Dupin and proper Dupin hypersurfaces in Rn or Sn can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective setting for the study of Dupin hypersurfaces, and many classifications of proper Dupin hypersurfaces have been obtained up to Lie sphere transformations. In these notes, we give a detailed introduction to this method for studying Dupin hypersurfaces in Rn or Sn, including proofs of several fundamental results. We also give a survey of the results in the field that have been obtained using this approach.
Reference55 articles.
1. Dupin hypersurfaces;Pinkall;Math. Ann.,1985
2. Über Komplexe, inbesondere Linien- und Kugelkomplexe, mit Anwendung auf der Theorie der partieller Differentialgleichungen;Lie;Math. Ann.,1872
3. Lie, S., and Scheffers, G. (1896). Geometrie der Berührungstransformationen, Teubner.
4. Cecil, T. (2008). Lie Sphere Geometry, with Applications to Submanifolds, Springer. [2nd ed.]. Universitext.
5. Smoothness theorems for the principal curvatures and principal vectors of a hypersurface;Singley;Rocky Mountain J. Math.,1975