Using Lie Sphere Geometry to Study Dupin Hypersurfaces in R^n

Authors

Thomas E. Cecil, College of the Holy CrossFollow

Document Type

Article

Publication Date

10-12-2021

Department

Mathematics

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.

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.

Comments

"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 arXiv.org 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. Arxiv.org 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.

DOI

https://doi.org/10.3390/axioms13060399

Published Article/Book Citation

Cecil, T.E. Using Lie Sphere Geometry to Study Dupin Hypersurfaces in Rn. Axioms 2024, 13, 399. https://doi.org/10.3390/axioms13060399

Repository Citation

Cecil, Thomas E., "Using Lie Sphere Geometry to Study Dupin Hypersurfaces in R^n" (2021). Mathematics and Computer Science Department Faculty Scholarship. 11.
https://crossworks.holycross.edu/math_fac_scholarship/11