THE TOPOLOGICAL SPHERE THEOREM FOR COMPLETE SUBMANIFOLDS

A topological sphere theorem is obtained from the point of view of sub manifold geometry. An important scalar is defined by the mean curvatur e and the squared norm of the second fundamental form of an oriented c omplete submanifold M-n in a space form of nonnegative sectional curva ture. If the infimum of this scalar is negative, we then prove that th e Ricci curvature of M-n has a positive lower bound, Making use of the Lawson-Simons formula for the nonexistence of stable k-currents, we e liminate H-k(M-n,Z) for all 1 < k < n - 1. We then observe that the fu ndamental group of M-n is trivial. It should be emphasized that our re sult is optimal.