The Topoligical Sphere Theorem for Complete Submanifolds

Abstract

A topological sphere theorem is obtained from the point of view of submanifold geometry. An important scalar is defined by the mean curvature and the squared norm of the second fundamental form of an oriented complete submanifold $M^n$ in a space form of nonnegative sectional curvature. If the infimum of this scalar is negative, we then prove that the 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 eliminate $H_k (M^n, {\bb Z})$ for all $1 < k < n - 1$. We then observe that the fundamental group of $M^n$ is trivial. It should be emphasized that our result is optimal.