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 rho > 0 and a fixed unit simple (n + 1)-vector a is an element of G(n+1,p)(p) such that the Gauss map g of an n-dimensional complete and connected spacelike submanifold M-n in S-p(n+p) satisfies <g, a> less than or equal to rho, then M-n is diffeomorphic to S-n, and its volume satisfies vol(S-n)/rho less than or equal to vol(M) less than or equal to rho(n)vol(S-n). We also characterize the case where these inequalities become equalities. (C) 2004 Elsevier B.V. All rights reserved.