## 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

### Citations:

Zbl 0359.53011; Zbl 0543.53046; Zbl 0443.53037
Full Text:

### References:

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