S Classification, With Three Different Principal Curvatures Of Moebius Isoparametric Hypersurfaces

Let Mⁿ be an immersed umbilic-free hypersurface in (n+1)-dimensional unit sphere Sⁿ⁺¹. According to Prof. Wang Changping's Mobius geometric theory of submanifolds, Mⁿ is associated with a so-called Mobius metric g, a Mobius second fundamental form B, a Blaschke tensor A and a M(o|¨)biusform Φ which are invariants of Mⁿ under the Mobius transformation group of Sⁿ⁺¹. A classical theorem of Mobius geometry states that Mⁿ(n ≥ 3) is in fact characterized by g and B up to Mobius equivalence. As a class of special hypersurfaces, a Mobius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ = 0, (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hypersurfaces are automatically Mobius isoparametrics, whereas the latter are Dupin hypersurfaces.In this paper, we study the Mobius isoparametric hypersurfaces in S⁶ with three distinct principal curvatures. Based on the classification theorem for hypersurfaces having parallel Mobius second fundamental form presented by Hu Zejun and Li Haizhong and that for hypersurfaces having two distinct Blaschke eigenvalues presented by Li Xingx-iao and Zhang Fengyun, we obtain the complete classification for Mobius isoparametric hypersurfaces in S⁶ with three distinct principle curatures.This paper develops those results established in recent years by Hu Zejun-Li Haizhong-Wang Changping et al. for Mobius isoparametric hypersurfaces in the unit sphere.