Classification of isoparametric hypersurfaces in spheres with (𝑔,𝑚)=(6,1)

Siffert, Anna

doi:10.1090/proc/12924

Classification of isoparametric hypersurfaces in spheres with $(g,m)=(6,1)$
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Proc. Amer. Math. Soc. 144 (2016), 2217-2230 Request permission

Abstract:

We classify the isospectral families $L(t)=\cos (t)L_0+\sin (t)L_1\in \mbox {Sym}(5,\mathbb {R})$, $t\in \mathbb {R}$, with $L_0=\mbox {diag}(\sqrt {3},\tfrac {1}{\sqrt {3}},0, -\tfrac {1}{\sqrt {3}},-\sqrt {3})$. Using this result we provide a classification of isoparametric hypersurfaces in spheres with $(g,m)=(6,1)$ and thereby give a simplified proof of the fact that any isoparametric hypersurface with $(g,m)=(6,1)$ is homogeneous. This result was first proven by Dorfmeister and Neher in 1985.

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