Nagoya Mathematical Journal
- Nagoya Math. J.
- Volume 204 (2011), 1-18.
Isoparametric hypersurfaces with four principal curvatures, II
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 in 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 is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs , , , and , 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 , , and remain open now.
Nagoya Math. J., Volume 204 (2011), 1-18.
First available in Project Euclid: 5 December 2011
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 53C40: Global submanifolds [See also 53B25]
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