Narrow Results

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 and connected submanifold M in S^n+p satisfies <g,a>≥ε, then M is diffeomorphic to Sⁿ, and the volume and diameter of M satisfy εⁿvol(Sⁿ)≤vol(M)≤vol(Sⁿ)/ε and 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.