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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
Citations:
Zbl 0578.53041
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References:
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