On the Work of Cartan and Münzner on Isoparametric Hypersurfaces

Authors

Thomas E. Cecil, College of the Holy CrossFollow

Document Type

Article

Publication Date

11-6-2024

Department

Mathematics

Abstract

A hypersurface Mⁿ in a real space form Rⁿ⁺¹, Sⁿ⁺¹, or Hⁿ⁺¹ is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan and Münzner on the theory of isoparametric hypersurfaces in real space forms, in particular, spheres. This work is contained in four papers of Cartan published during the period 1938--1940, and two papers of Münzner that were published in preprint form in the early 1970's, and as journal articles in 1980--1981. These papers of Cartan and Münzner have been the foundation of the extensive field of isoparametric hypersurfaces, and they have all been recently translated into English by the author. The paper concludes with a brief survey of the recently completed classification of isoparametric hypersurfaces in spheres.

Comments

Subjects: Differential Geometry (math.DG)
MSC classes: 53C40 (Primary) 53C42, 53B25 (Secondary)
Cite as: arXiv:2411.04231 [math.DG]
(or arXiv:2411.04231v1 [math.DG] for this version)

DOI

https://doi.org/10.48550/arXiv.2411.04231

Creative Commons License

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

Repository Citation

Cecil, Thomas E., "On the Work of Cartan and Münzner on Isoparametric Hypersurfaces" (2024). Mathematics and Computer Science Department Faculty Scholarship. 21.
https://crossworks.holycross.edu/math_fac_scholarship/21