A new proof of the homogeneity of isoparametric hypersurfaces with \((g,m)=(6,1)\) (Q2735671)

By a result of \textit{H. F. Münzner}, the number \(g\) of different multiplicities an isoparametric hypersurface \(M\) in a sphere can have is \(1\), \(2\), \(3\), \(4\) or \(6\) [Math. Ann. 251, 57-71 (1980; Zbl 0417.53030) and Math. Ann. 256, 215-232 (1981; Zbl 0438.53050)]. If \(g=1\), \(2\) or \(3\), then \textit{E. Cartan} proved that \(M\) is homogeneous [Ann. Mat. Pura Appl., IV. Ser. 17, 177-191 (1938; Zbl 0020.06505) and Math. Z. 45, 335-367 (1939; Zbl 0021.15603)]. If \(g=4\), then inhomogeneous examples were found by \textit{H. Ozeki} and \textit{M. Takeuchi} [Tôhoku Math. J. 28, 7-55 (1976; Zbl 0359.53012)] and later in a more systematic way by \textit{D. Ferus, H. Karcher} and \textit{H.-F. Münzner} [Math. Z. 177, 479-502 (1981; Zbl 0443.53037)].NEWLINENEWLINENEWLINEIn the paper under review the last case \(g=6\) is dealt with. \textit{U. Abresch} had already proved in [Math. Ann. 264, 283-302 (1983; Zbl 0514.53050)] that the multiplicities of the principal curvatures in this case are either all equal to \(1\) or all equal to \(2\). Using an algebraic approach, \textit{J. Dorfmeister} and \textit{E. Neher} proved in [Commun. Algebra 13, 2299-2368 (1985; Zbl 0578.53041)] that \(M\) is homogeneous if \(g=6\) and the multiplicity \(m=1\).NEWLINENEWLINENEWLINEThis theorem is reproved in the paper under review using a more geometric approach. The author refers to a preprint of hers from the year 2000, ``Homogeneity of isoparametric hypersurfaces with six principal curvatures'', in which the method of the present paper is applied to isoparametric hypersurfaces in spheres with \(g=6\) and \(m=2\) to prove that they are homogeneous. In spite of the existence of this preprint, the case \(g=6\) and \(m=2\) is generally considered to be open.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00038].