Classification of Möbius Isoparametric Hypersurfaces in ${\Bbb S}^5$

Abstract

Let Mn be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere \({\Bbb S}^{n+1}\), then Mn is associated with a so-called Mobius metric g, a Mobius second fundamental form B and a Mobius form Φ which are invariants of Mn under the Mobius transformation group of \({\Bbb S}^{n+1}\). A classical theorem of Mobius geometry states that Mn (n ≥ 3) is in fact characterized by g and B up to Mobius equivalence. A Mobius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hypersurfaces are automatically Mobius isoparametrics, whereas the latter are Dupin hypersurfaces.