Connectedness of the algebraic set of vectors generating planar normal sections of homogeneous isoparametric hypersurfaces

摘要

Let $Msubset mathbb{S}^{n+1}subset $ $mathbb{R}^{n+2}$ be a homogeneous isoparametric hypersurface and consider the algebraic set of unit tangent vectors generating planar normal sections at a point $Ein M$ (denoted by $widehat{X}_{E}[M] subset T_{E}(M)$). The present paper is devoted to prove that $widehat{X}_{E}[M]$ is connected by arcs . This in turn proves that its projective image $X[M] subset mathbb{RP}(T_{E}(M))$ also has this property.