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December 2011 Isoparametric hypersurfaces with four principal curvatures, II
Quo-Shin Chi
Nagoya Math. J. 204: 1-18 (December 2011). DOI: 10.1215/00277630-1431813


In this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities (3,4) in S15 is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint m22m11 is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs (4,5), (3,4), (7,8), and (6,9), where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra, and the complexified octonion algebra, whereas the first stands alone in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs (4,5), (6,9), and (7,8) remain open now.


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Quo-Shin Chi. "Isoparametric hypersurfaces with four principal curvatures, II." Nagoya Math. J. 204 1 - 18, December 2011.


Published: December 2011
First available in Project Euclid: 5 December 2011

zbMATH: 1243.53094
MathSciNet: MR2863363
Digital Object Identifier: 10.1215/00277630-1431813

Primary: 53C40

Rights: Copyright © 2011 Editorial Board, Nagoya Mathematical Journal


Vol.204 • December 2011
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