Osaka Journal of Mathematics

The Dorfmeister-Neher theorem on isoparametric hypersurfaces

Reiko Miyaoka

Full-text: Open access


A new proof of the homogeneity of isoparametric hypersurfaces with six simple principal curvatures [4] is given in a method applicable to the multiplicity two case.

Article information

Osaka J. Math., Volume 46, Number 3 (2009), 695-715.

First available in Project Euclid: 26 October 2009

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C40: Global submanifolds [See also 53B25]


Miyaoka, Reiko. The Dorfmeister-Neher theorem on isoparametric hypersurfaces. Osaka J. Math. 46 (2009), no. 3, 695--715.

Export citation


  • U. Abresch: Isoparametric hypersurfaces with four or six distinct principal curvatures. Necessary conditions on the multiplicities, Math. Ann. 264 (1983), 283--302.
  • É. Cartan: Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl. 17 (1938), 177--191.
  • T.E. Cecil, Q.-S. Chi and G.R. Jensen: Isoparametric hypersurfaces with four principal curvatures, Ann. of Math. (2) 166 (2007), 1--76.
  • J. Dorfmeister and E. Neher: Isoparametric hypersurfaces, case $g=6$, $m=1$, Comm. Algebra 13 (1985), 2299--2368.
  • S. Immervoll: On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres, Ann. of Math. (2) 168 (2008), 1011--1024.
  • R. Miyaoka: The linear isotropy group of $G\sb 2/\textit{SO}(4)$, the Hopf fibering and isoparametric hypersurfaces, Osaka J. Math. 30 (1993), 179--202.
  • R. Miyaoka: A new proof of the homogeneity of isoparametric hypersurfaces with $(g,m)=(6,1)$; in Geometry and Topology of Submanifolds, X (Beijing/Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 178--199.
  • R. Miyaoka: Isoparametric geometry and related topics; in Surveys on Geometry and Integrable Systems, Adv. Stud. Pure Math. 51, Math. Soc. Japan, Tokyo, 2008, 315--337.
  • R. Miyaoka: Isoparametric hypersurfaces with $(g,m)=(6,2)$, preprint (2009).
  • H.F. Münzner: Isoparametrische Hyperflächen in Sphären, I, Math. Ann. 251 (1980), 57--71.
  • G. Thorbergsson: A survey on isoparametric hypersurfaces and their generalizations; in Handbook of Differential Geometry, I, North-Holland, Amsterdam, 2000, 963--995.