Mobius Isoparametric Hypersurfaces in S^(n+1) with Two Distinct Principal Curvatures

A hypersurface x: M→S^(n+1) without umbilic point is called a Mbius isoparametric hypersurface if its Mbius form Φ=-ρ^(-2)∑_i(ei(H)+∑_j(h_(ij)-Hδ_(ij))e_j(logρ))θ_i vanishes and its Mbius shape operator S=ρ^(-1)(S-Hid) has constant eigenvalues. Here {e_i} is a local orthonormal basis for I=dx·dx with dual basis {θ_i}, II=∑_(ij)h_(ij)θ_iθ_J is the second fundamental form, H=1/n∑_i h_(ij), ρ~2=n/(n-1)(‖II‖~2-nH^2) and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S^(n+1) is a Mbius isoparametric hypersurface, but the converse is not true. In this paper we classify all Mbius isoparametric hypersurfaces in S^(n+1) with two 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 isoparametric hypersurface embedded in S^(n+1) can take only the values 2, 3, 4, 6.