Abstract
We consider Mn, n ≥ 3, an n-dimensional complete submanifold of a Riemannian manifold $(\overline{M}^{n+p},\overline{g})$. We prove that if for all point x ∈ Mn the following inequality is satisfied
$$S\leq\frac{8}{3} \bigg( \overline{K}_{\min}-\frac{1}{4}\overline{K}_{\max} \bigg)+\frac{n^2H^2}{n-1},$$
with strictly inequality at one point, where S and H denote the squared norm of the second fundamental form and the mean curvature of Mn respectively, then Mn is either diffeomorphic to a spherical space form or the Euclidean space ℝn. In particular, if Mn is simply connected, then Mn is either diffeomorphic to the sphere $\mathbb{S}$n or the Euclidean space ℝn.
