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 ℝ
n. In particular, if
Mn is simply connected, then
Mn is either diffeomorphic to the sphere $\mathbb{S}$
n or the Euclidean space ℝ
n.
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