#### Document Type

Article

#### Publication Date

11-20-2020

#### Department

Mathematics

#### Abstract

In these notes, we give a detailed account of the method for studying Dupin hypersurfaces in * R^{n}* or

**using Lie sphere geometry, and we conclude with a classification of the cyclides of Dupin obtained by using this approach.**

*S*^{n}Specifically, an oriented hypersurface

**⊂**

*M*^{n−1}*is a cyclide of Dupin of characteristic (*

**R**^{n}*p,q*), where

*p*+

*q*=

*n*− 1, if

**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 (**

*M*^{n−1}*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

*(1 / √2) ×*

**S**^{q}*(1 / √2) ⊂*

**S**^{p}*. As a corollary, we also derive a Möbius geometric classification of the cyclides in*

**S**^{n}*.*

**R**^{n}#### Repository Citation

Cecil, Thomas E., "Notes on Lie Sphere Geometry and the Cyclides of Dupin" (2020). *Mathematics Department Faculty Scholarship*. 11.

https://crossworks.holycross.edu/math_fac_scholarship/11

## Comments

Link to ArXiv version: arXiv:2011.11432