×

A new look at condition A. (English) Zbl 1246.53078

An isoparametric hypersurface in \(S^n\) can be obtained as the intersection of \(S^n\) with the level set of a homogeneous polynomial on \(\mathbb {R}^{n+1}\), of degree \(g=1\), \(2\), \(3\), \(4\) or \(6\). These isoparametric hypersurfaces have \(g\) principal curvatures, where only \(2\) different multiplicities \(m_i\), \(i=1,2\), of the principal curvatures occur. For the construction of inhomogeneous isoparametric hypersurfaces with \(4\) different principal curvatures, H. Ozeki and M. Takeuchi [Tohoku Math. J., II. Ser. 27, 515–559 (1975; Zbl 0359.53011)] imposed conditions (“A” and “B”) on the focal submanifolds of the isoparametric hypersurface, the first of which appeared to be crucial in subsequent work. In particular, J. Dorfmeister and E. Neher [Osaka J. Math. 20, 145–175 (1983; Zbl 0543.53046)] showed that the first of these conditions suffices to characterize a particular class of isoparametric hypersurfaces [D. Ferus, H. Karcher and H.-F. Münzner, Math. Z. 177, 479–502 (1981; Zbl 0443.53037)]. The author provides a simpler and more geometric proof of this result, aiming to shed light on the geometric meaning of the condition for the construction of classes of examples, paying particular attention to the exceptional multiplicity pairs \((3,4)\) and \((7,8)\). In particular, the author analyses a construction of isoparametric hypersurfaces using octonians in order to provide a better understanding of the relation between the algebraic and geometric structures involved.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C40 Global submanifolds
PDF BibTeX XML Cite
Full Text: arXiv Euclid

References:

[1] E. Cartan: Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques , Math. Z. 45 (1939), 335-367. · Zbl 0021.15603
[2] T.E. Cecil, Q.-S. Chi and G.R. Jensen: Isoparametric hypersurfaces with four principal curvatures , Ann. of Math. (2) 166 (2007), 1-76. · Zbl 1143.53058
[3] J. Dorfmeister and E. Neher: An algebraic approach to isoparametric hypersurfaces in spheres , I, Tôhoku Math. J. (2) 35 (1983), 187-224. · Zbl 0507.53038
[4] J. Dorfmeister and E. Neher: An algebraic approach to isoparametric hypersurfaces in spheres , II, Tôhoku Math. J. (2) 35 (1983), 225-247. · Zbl 0507.53039
[5] J. Dorfmeister and E. Neher: Isoparametric triple systems of algebra type , Osaka J. Math. 20 (1983), 145-175. · Zbl 0543.53046
[6] J. Dorfmeister and E. Neher: Isoparametric triple systems of FKM-type , II, Manuscripta Math. 43 (1983), 13-44. · Zbl 0531.53049
[7] D. Ferus, H. Karcher and H.F. Münzner: Cliffordalgebren und neue isoparametrische Hyperflächen , Math. Z. 177 (1981), 479-502. · Zbl 0443.53037
[8] D. Husemoller: Fibre Bundles, third edition, Graduate Texts in Mathematics 20 , Springer, New York, 1994. · Zbl 0202.22903
[9] K. McCrimmon: Quadratic forms permitting triple composition , Trans. Amer. Math. Soc. 275 (1983), 107-130. · Zbl 0505.17004
[10] R. Miyaoka: The Dorfmeister-Neher theorem on isoparametric hypersurfaces , Osaka J. Math. 46 (2009), 695-715. · Zbl 1185.53059
[11] R. Miyaoka: Isoparametric hypersurfaces with \((g,m)=(6,2)\) , · Zbl 1263.53049
[12] H.F. Münzner: Isoparametrische Hyperflächen in Sphären , Math. Ann. 251 (1980), 57-71. · Zbl 0417.53030
[13] H.F. Münzner: Isoparametrische Hyperflächen in Sphären , II, Math. Ann. 256 (1981), 215-232. · Zbl 0438.53050
[14] H. Ozeki and M. Takeuchi: On some types of isoparametric hypersurfaces in spheres , I, Tôhoku Math. J. (2) 27 (1975), 515-559. · Zbl 0359.53011
[15] \begingroup H. Ozeki and M. Takeuchi: On some types of isoparametric hypersurfaces in spheres , II, Tôhoku Math. J. (2) 28 (1976), 7-55. \endgroup · Zbl 0359.53012
[16] S. Stolz: Multiplicities of Dupin hypersurfaces , Invent. Math. 138 (1999), 253-279. · Zbl 0944.53035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.