Möbius Isoparametric Hypersurfaces in S n +1 with Two Distinct Principal Curvatures

Abstract

A hypersurface x : M → S n+1 without umbilic point is called a Mobius isoparametric hypersurface if its Mobius form Φ = −ρ −2∑ i (e i (H) + ∑ j (h ij −Hδ ij )e j (log ρ))θ i vanishes and its Mobius shape operator $$ {\Bbb {S}}$$ = 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 i ⊗θ i is the second fundamental form, $$H = {1 \over n}\sum\nolimits_i {h_{ii} ,\rho^2 = {n \over {n - 1}}( {|| II ||^2 - nH^2 } )}$$ 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 Mobius isoparametric hypersurface, but the converse is not true. In this paper we classify all Mobius isoparametric hypersurfaces in S n+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 S n+1 can take only the values 2, 3, 4, 6.