I have the following exercise which I cannot solve:
Show that there exists a smooth map $\mathbb{R}^m \mapsto \mathcal{S}^m$ onto the m-sphere such that the open ball $\{x \in \mathbb{R}^m | \: \| x\|_{2} < 1 \}$ is mapped diffeomorphically onto the complement $\mathcal{S}^m \setminus \{s\}$ of a point $s \in \mathcal{S}^m$ and its exterior $\{x \in \mathbb{R}^m | \: \|x\|_{2} \geq 1 \}$ is mapped constantly to s.
If anyone has an idea for that part that would be really helpful.
What I already tried: We can argue with symmetry so it's obvious we only need to construct such a map for s equal the south pole of the sphere. I think the stereographic-projection should work as a diffeomorphism from $\mathbb{R}^m \mapsto \mathcal{S}\setminus \{s\}$. But it does not map the exterior $\{x \in \mathbb{R}^m | \: \|x\|_{2} \geq 1 \}$ entirely to the point s.
Thanks in advance !