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

A hypersurface x : M → Sn+1 without umbilic point is called a Möbius isoparametric hypersurface if its Möbius form Φ = −ρ−2∑i(ei(H) + ∑j(hij−Hδij)ej(log ρ))θi vanishes and its Möbius shape operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\Bbb {S}}$$\end{document} = ρ−1(S−Hid) has constant eigenvalues. Here {ei} is a local orthonormal basis for I = dx·dx with dual basis {θi}, II = ∑ijhijθi⊗θi is the second fundamental form, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H = {1 \over n}\sum\nolimits_i {h_{ii} ,\rho^2 = {n \over {n - 1}}( {|| II ||^2 - nH^2 } )}$$\end{document} 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 Möbius isoparametric hypersurface, but the converse is not true. In this paper we classify all Möbius isoparametric hypersurfaces in Sn+1 with two distinct principal curvatures up to Möbius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact Möbius isoparametric hypersurface embedded in Sn+1 can take only the values 2, 3, 4, 6.