单位球面上具有非负M bius截面曲率的超曲面

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~ .

The Hypersurfaces in a Unit Sphere with Nonnegative M bius Sectional Curvature