Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 24H (Differential Geometry)

At a point where $df$ is surjective, choose charts and reorder the domain coordinates so that an $\ell \times \ell$ minor of the derivative is invertible. The map $F=(f_1,\dots ,f_{\ell },x_{\ell +1},\dots ,x_k)$ has invertible derivative. By the inverse function theorem it gives local coordinates, in which $f$ is the projection $(u_1,\dots ,u_k)\mapsto (u_1,\dots ,u_{\ell })$. This is the submersion theorem.

Coordinate projections are open maps, so the local form of the submersion proves that $f$ is open. If $X$ is compact and nonempty, its continuous image is compact and hence closed in the Hausdorff manifold $Y$. Therefore $f(X)$ is a nonempty set both open and closed in connected $Y$, giving $f(X)=Y$.

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.