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Isoparametric hypersurfaces with $$(g,m)=(6,2)$$. (English) Zbl 1263.53049
Ann. Math. (2) 177, No. 1, 53-110 (2013); erratum ibid. (2) 183, No. 3, 1057-1071 (2016).
A hypersurface of a space form $$M$$ is said to be isoparametric when it has constant principal curvatures. Such surfaces were studied already by Elli Cartan. The most interesting case is $$M = S^n$$. It is known that the number $$g$$ of different principal curvatures of such hypersurfaces belongs to the set $$\{ 1, 2, 3, 4, 6\}$$ and that if $$g = 6$$, then the multiplicities $$m$$ of all principal curvatures are the same and equal either $$1$$ or to $$2$$. In this remarkable paper, the author proves the following:
{ Theorem 1.1.} All the isoparametric hypersurfaces of the spheres with $$(g, m) = (6,2)$$ are homogeneous.
The similar result for $$(g, m) = (6,1)$$ has been proven by J. Dorfmeister and E. Neher [Commun. Algebra 13, 2299–2368 (1985; Zbl 0578.53041)]. The author of this paper worked with isoparametric surfaces for several years and produced a number of important results in this area. A reader seriously interested in this topic is adviced to read also earlier papers by R. Miyaoka.

##### MSC:
 53C40 Global submanifolds 53A05 Surfaces in Euclidean and related spaces
##### Keywords:
isoparametric hypersurfaces; homogenous spaces
Zbl 0578.53041
Full Text:
##### References:
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