 Solution

A submersion from a nonempty compact manifold to ${\mathbb{R}}^{\ell }$, $\ell \ge 1$, would be surjective by the preceding result, since the target is connected. Its image would also be compact, contradicting noncompactness of ${\mathbb{R}}^{\ell }$. Thus no such submersion exists. The nonempty-domain qualification excludes the vacuous empty-manifold case.