Document Type


Publication Date





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.

NOTE: This paper is a revised version of "Notes on Lie Sphere Geometry and the Cyclides of Dupin" and is published as such despite having a different title than the original paper.


"Using Lie Sphere Geometry to Study Dupin Hypersurfaces in Rn" is a revision (with two new sections) of an earlier paper entitled "Notes on Lie Sphere Geometry and the Cyclides of Dupin." It was submitted to as a revision of "Notes on Lie Sphere Geometry and the Cyclides of Dupin" because the author determined there was too much overlap with "Notes on Lie Sphere Geometry and the Cyclides of Dupin," for it to be considered to be a new paper. accepted it as such and replaced the earlier paper, even though the new version has a different title.

Link to ArXiv version: arXiv:2011.11432v2

The original paper is included as a supplementary file.

cyclides.pdf (316 kB)
Notes on Lie Sphere Geometry and the Cyclides of Dupin


To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.