Classifications of Dupin Hypersurfaces in Lie Sphere Geometry

Authors

Thomas E. Cecil, College of the Holy CrossFollow

Document Type

Article

Publication Date

10-19-2022

Department

Mathematics

Abstract

This is a survey of local and global classification results concerning Dupin hypersurfaces in Sⁿ (or Rⁿ) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres. Along with these classification results, many important concepts from Lie sphere geometry, such as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of Sⁿ (or Rⁿ), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.

Comments

Link to ArXiv version: https://arxiv.org/abs/2210.10569

AMS Classification numbers: 53A07, 53A40, 53B25, 53C40, 53C42

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Published Article/Book Citation

Cecil, T.E. Classifications of Dupin hypersurfaces in Lie sphere geometry. Acta Math Sci 44, 1–36 (2024). https://doi.org/10.1007/s10473-024-0101-7

Repository Citation

Cecil, Thomas E., "Classifications of Dupin Hypersurfaces in Lie Sphere Geometry" (2022). Mathematics and Computer Science Department Faculty Scholarship. 16.
https://crossworks.holycross.edu/math_fac_scholarship/16