The Hypersurfaces in a Unit Sphere with Nonnegative Mobius Sectional Curvature

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 M(?)bius invariants of x under the M(?)bius transformation group of S^(n+1) are M(?)bius metric,M(?)bius form,M(?)bius 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 M(?)bius sectional curvature and with vanishing M(?)bius form.Then x is locally M(?)bius equivalent to one of the following hypersurfaces:(i)the torus S^k(a)×S^(n-k)((1-a^2)^(1/2))with 1≤k≤n-1;(ii)the pre-image of the stereographic projection of the standard cylinder S^k×R^(n-k)(?)R^(n+1) with 1≤k≤n-1;(iii)the pre-image of the stereographic projection of the cone in R^(n+1):(?)(u,v,t)=(tu,tv), where(u,v,t)∈S^k(a)×S^(n-k-1)((1-a^2)^(1/2))×R^+.