On M(?)bius Isoparametric Hypersurfaces In S～7

Let x: Mⁿ→Sⁿ⁺¹ be an immersed umbilic-free hypersurface in the (n+1)-dimensional unite sphere Sⁿ⁺¹. According to Wang Changping's Mo|¨bius geometric theory of subman-ifolds. Mⁿ is associated with a so-called Mo|¨bius metric g, a Mo|¨bius second fundamental form B, a Blaschke tensor A and a Mo|¨bius formΦwhich are invariants of Mⁿ under the Mo|¨bius transformation group of Sⁿ⁺¹. a Mo|¨bius isoparametric hypersurface is defined by satisfying the conditionsΦ= 0 and that all the eigenvalues of B with respect to g are constants. Now. Mo|¨bius isoparametric hypersurfaces in Sⁿ⁺¹(n≤5) have been completely classified. In this paper, we study the Mo|¨bius isoparametric hypersurfaces in S7, and in the case that the hypersurfaces with three or four distinct principal curvatures, we can obtain the following classification theorem:Main classification theorem. Let x : M⁶→S⁷ be a Mo|¨bius isoparametric hypersurface with three or four distinct Mo|¨bius principal curvatures , Then x is Mo|¨bius equivalent to an open part of one of the following hypersurfaces in S⁷:(1) The preimage of the stereo-graphic projection of the warped product embeddingwith p≥1, q≥1: p + q≤5, 0 < a < 1, defined byand the multiplicity of the principal curvatures is p, q, 6—p—q, respectively.(2) Minimal hypersurfaces defined bywith here, y₁ : N³â†’S⁴((36)/5)^1/2(?)R⁵ is E. Cartan's minimal isoparametric hypersurface with vanishing scalar curvature and principal curvatures (5/(12))^1/2,0, -(5/(12))^1/2; and (y₀;y₁) :H³(-5/(36)) (?)L⁴ is the standard embedding of the hyperbolic space of sectional curvature -5/(36) into the 4- dimensional Lorentz space with -y₀²+y₂² = - (36)/5, and the multiplicity of the principal curvatures is 1, 1, 4. respectively.(3) an Euclidean isoparametric hypersurface with three distinct principal curvatures inS⁷, and they have the same multiplicity two.(4) an Euclidean isoparametric hypersurface with four distinct principal curvatures in S⁷,and the multiplicity of the principal curvatures is 1, 1, 2, 2, respectively.