Homogeneous spaces and isoparametric hypersurfaces

Stefan Immervoll

Abstract: \def\eset \font\callmittel=eufm8 \def\hmittel{{\hbox \hmitel}} \def\hmitel{{\callmittel h}} In this note we establish a relation between isoparametric hypersurfaces with four distinct principal curvatures in spheres and homogeneous spaces. Let $m_1, m_2$ denote the multiplicities of the principal curvatures. Then the orbit $N(m_1, m_2) = \eset{A \in \hmittel(2m_1 +m_2)}{{\rm tr}(A)=0,\, {\rm rank} (A) = 2 m_1,\,\hbox{and}\, A^3 = A}$ of the action of the orthogonal group $Ø(2m_1+m_2)$ on the real symmetric matrices $\scriptstyle\hmittel(2m_1+m_2)$ contains a totally geodesic, $m_2$-dimensional round sphere. Here $N(m_1, m_2)$ is endowed with the metric induced by a scalar product on $\hmittel(2m_1+m_2)$ defined by the trace.

Classification (MSC2000): 53C30; 53C40, 17A40

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