On the Work of Cartan and M\"{u}nzner on Isoparametric Hypersurfaces

TL;DR

This paper surveys the foundational work of Cartan and M"{u}nzner on isoparametric hypersurfaces in real space forms, highlighting their contributions and recent classification results, especially in spheres.

Contribution

It provides a comprehensive overview of the historical and recent developments in the theory of isoparametric hypersurfaces, emphasizing the foundational work of Cartan and M"{u}nzner.

Findings

Cartan's work from 1938-1940 laid the groundwork for the field.

M"{u}nzner's 1980-1981 papers advanced the classification of isoparametric hypersurfaces.

Recent classification results have completed the understanding of isoparametric hypersurfaces in spheres.

Abstract

A hypersurface $M^n$ in a real space form ${\bf R}^{n+1}$, $S^{n+1}$, or $H^{n+1}$ is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan and M\"{u}nzner 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 M\"{u}nzner that were published in preprint form in the early 1970's, and as journal articles in 1980--1981. These papers of Cartan and M\"{u}nzner 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.