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 Mⁿ in a space form of nonnegative sectional curvature. If the infimum of this scalar is negative, we then prove that the Ricci curvature of Mⁿ has a positive lower bound. Making use of the Lawson–Simons formula for the nonexistence of stable k -currents, we eliminate Hₖ (Mⁿ, Z) for all $1 < k < n - 1$ . We then observe that the fundamental group of Mⁿ is trivial. It should be emphasized that our result is optimal.
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Shiohama et al. (1997) studied this question.