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28 pages, 399 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Work of Cartan and Münzner on Isoparametric Hypersurfaces

by Thomas E. Cecil and Patrick J. Ryan

Axioms 2025, 14(1), 56; https://doi.org/10.3390/axioms14010056 - 13 Jan 2025

Cited by 1 | Viewed by 901

A hypersurface

M^{n}

in a real space form

R^{n + 1}

S^{n + 1}

, or

H^{n + 1}

is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan [...] Read more.

A hypersurface

M^{n}

in a real space form

R^{n + 1}

S^{n + 1}

, or

H^{n + 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 1970s 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 T. Cecil. The paper concludes with a brief survey of the recently completed classification of isoparametric hypersurfaces in spheres. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

36 pages, 446 KB

Open AccessFeature PaperArticle

Using Lie Sphere Geometry to Study Dupin Hypersurfaces in Rⁿ

by Thomas E. Cecil

Axioms 2024, 13(6), 399; https://doi.org/10.3390/axioms13060399 - 14 Jun 2024

Cited by 1 | Viewed by 1334

Abstract

A hypersurface M in

R^{n}

S^{n}

A hypersurface M in

R^{n}

S^{n}

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

R^{n}

S^{n}

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

R^{n}

S^{n}

, including proofs of several fundamental results. We also give a survey of the results in the field that have been obtained using this approach. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)

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