Mobius Isoparametric Hypersurfaces in Sn+1 with Two Distinct Principal Curvatures

<正> A hypersurface x: M→Sn+1 without umbilic point is called a Mbius isoparametrichypersurface if its Mbius form Φ=-ρ-2∑i(ei（H）+∑j(hij-Hδij)ej（logρ）)θi vanishes and itsMbius shape operator S=ρ-1（S-Hid） has constant eigenvalues. Here {ei} is a local orthonormalbasis for I=dx·dx with dual basis {θi}, II=∑ijhijθiθJ is the second fundamental form,H=1/n∑i hij, ρ2=n/（n-1）（‖II‖2-nH2） and S is the shape operator of x. It is clear that any conformalimage of a （Euclidean） isoparametric hypersurface in Sn+1 is a Mbius isoparametric hypersurface, butthe converse is not true. In this paper we classify all Mbius isoparametric hypersurfaces in Sn+1 withtwo distinct principal curvatures up to Mbius transformations. By using a theorem of Thorbergsson[1] we also show that the number of distinct principal curvatures of a compact Mbius isoparametrichypersurface embedded in Sn+1 can take only the values 2, 3, 4, 6.