In these notes, we give a detailed account of the method for studying Dupin hypersurfaces in Rn or Sn using Lie sphere geometry, and we conclude with a classification of the cyclides of Dupin obtained by using this approach.
Specifically, an oriented hypersurface Mn−1⊂Rn is a cyclide of Dupin of characteristic (p,q), where p +q = n − 1, if Mn−1 has two distinct principal curvatures at each point with respective multiplicities p and q, and each principal curvature function is constant along each leaf of its corresponding principal foliation. We show that every connected cyclide of characteristic (p,q) is contained in a unique compact, connected cyclide of characteristic (p,q). Furthermore, every compact, connected cyclide of characteristic (p,q) is equivalent by a Lie sphere transformation to a standard product of two spheres Sq(1 / √2) × Sp(1 / √2) ⊂ Sn. As a corollary, we also derive a Möbius geometric classification of the cyclides in Rn.
Cecil, Thomas E., "Notes on Lie Sphere Geometry and the Cyclides of Dupin" (2020). Mathematics Department Faculty Scholarship. 11.