A submersion $f: S^2 \rightarrow \mathbb{R}^2$

Solution 1:

Note that a submersion between two manifolds of equal dimension is a local diffeomorphism (since a linear map $\mathbb{F}^n \to \mathbb{F}^n$ is surjective iff it's injective, for any field $\mathbb{F}$ and any $n \in \mathbb{N}$).

Therefore, the map $S^2 \to \mathbb{R}^2$ is an open map. Its image must therefore be an open set. The image must also be compact, hence closed and bounded. Therefore, the image is a nonempty clopen proper subset of $\mathbb{R}^2$. But this contradicts the fact that $\mathbb{R}^2$ is connected.

So there is no submersion $S^2 \to \mathbb{R}^2$.

Solution 2:

If the tangent map was everywhere surjective, then it would also be everywhere injective. Then, we could pull back the distinguished vector field in the $x$ direction to produce a non-vanishing vector field on the sphere. This would contradict the hairy ball theorem.