A hypersurface x: M → Sn+1 without umbilic point is called a Mobius isoparametric hypersurface if its Mobius form Φ=−ρ−2 Σi(ei(H) + Σj (hij−Hδij)ej(log ρ))θi vanishes and its Mobius shape operator S=ρ−1(S−Hid) has constant eigenvalues. Here {ei} is a local orthonormal basis for I=dx·dx with dual basis {θi}, II =Σ ij hijθi ⊗ θj is the second fundamental form, H=1 n Σi hii, ρ2=n n−1 (||II||2−nH2) and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in Sn+1 is a Mobius isoparametric hypersurface, but the converse is not true. In this paper we classify all Mobius isoparametric hypersurfaces in Sn+1 with two distinct principal curvatures up to Mobius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact Mobius isoparametric hypersurface embedded in Sn+1 can take only the values 2, 3, 4, 6.

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