ON THE TOPOLOGY,VOLUME,DIAMETER AND GAUSS MAP IMAGE OF SUBMANIFOLDS IN A SPHERE

In this paper, the author uses Gauss map to study the topology, volume and diameter of submanifolds in a sphere. It is proved that if there exist ε, 1 ≥ε≥ 0 and a fixed unit simple p-vector a such that the Gauss map g of an n-dimensional complete to Sn, and the volume and diameter of M satisfy εnvol(Sn) ≤vol(M) ≤ vol(Sn)/εand επ≤diam(M) ≤π/ε, respectively. The author also characterizes the case where these inequalities become equalities. As an application, a differential sphere theorem for compact submanifolds in a sphere is obtained.