摘要
Let x : M→S^n+1 be a hypersurface in the (n + 1)-dimensional unit sphere S^n+1 without umbilic point. The Mobius invariants of x under the Mobius transformation group of S^n+1 are Mobius metric, Mobius form, Mobius second fundamental form and Blaschke tensor. In this paper, we prove the following theorem: Let x : M→S^n+1 (n≥2) be an umbilic free hypersurface in S^n+1 with nonnegative Mobius sectional curvature and with vanishing Mobius form. Then x is locally Mobius equivalent to one of the following hypersurfaces: (i) the torus S^k(a) × S^n-k(√1- a^2) with 1 ≤ k ≤ n - 1; (ii) the pre-image of the stereographic projection of the standard cylinder S^k × R^n-k belong to R^n+1 with 1 ≤ k ≤ n- 1; (iii) the pre-image of the stereographic projection of the Cone in R^n+1 : -↑x(u, v, t) = (tu, tv), where (u,v, t)∈S^k(a) × S^n-k-1( √1-a^2)× R^+.
Let x : M→S^n+1 be a hypersurface in the (n + 1)-dimensional unit sphere S^n+1 without umbilic point. The Mobius invariants of x under the Mobius transformation group of S^n+1 are Mobius metric, Mobius form, Mobius second fundamental form and Blaschke tensor. In this paper, we prove the following theorem: Let x : M→S^n+1 (n≥2) be an umbilic free hypersurface in S^n+1 with nonnegative Mobius sectional curvature and with vanishing Mobius form. Then x is locally Mobius equivalent to one of the following hypersurfaces: (i) the torus S^k(a) × S^n-k(√1- a^2) with 1 ≤ k ≤ n - 1; (ii) the pre-image of the stereographic projection of the standard cylinder S^k × R^n-k belong to R^n+1 with 1 ≤ k ≤ n- 1; (iii) the pre-image of the stereographic projection of the Cone in R^n+1 : -↑x(u, v, t) = (tu, tv), where (u,v, t)∈S^k(a) × S^n-k-1( √1-a^2)× R^+.
基金
Foundation item:The NNSF(10671087)of China
the NNSF(0511008)of Jiangxi Province,China.