On the volume and Gauss map image of spacelike submanifolds in de Sitter space form

In this paper we use Gauss map to study spacelike submanifolds in de Sitter space form. We prove that if there exist ρ>0 and a fixed unit simple (n+1)-vector a∈Gn+1,pp such that the Gauss map g of an n-dimensional complete and connected spacelike submanifold Mn in Spn+p satisfies 〈g,a〉≤ρ, then Mn is diffeomorphic to Sn, and its volume satisfies vol(Sn)/ρ≤vol(M)≤ρn vol(Sn). We also characterize the case where these inequalities become equalities.