Nagoya Mathematical Journal

Isoparametric hypersurfaces with four principal curvatures, II

Quo-Shin Chi

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Abstract

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.

Article information

Source
Nagoya Math. J., Volume 204 (2011), 1-18.

Dates
First available in Project Euclid: 5 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1323107835

Digital Object Identifier
doi:10.1215/00277630-1431813

Mathematical Reviews number (MathSciNet)
MR2863363

Zentralblatt MATH identifier
1243.53094

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

Citation

Chi, Quo-Shin. Isoparametric hypersurfaces with four principal curvatures, II. Nagoya Math. J. 204 (2011), 1--18. doi:10.1215/00277630-1431813. https://projecteuclid.org/euclid.nmj/1323107835


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