Let $f: X \rightarrow Y$ be a smooth map between manifolds without boundary. Recall that $f$ is a submersion if $d f_{x}: T_{x} X \rightarrow T_{f(x)} Y$ is surjective for all $x \in X$. The canonical submersion is the standard projection of $\mathbb{R}^{k}$ onto $\mathbb{R}^{l}$ for $k \geqslant l$, given by

$$\left(x_{1}, \ldots, x_{k}\right) \mapsto\left(x_{1}, \ldots, x_{l}\right)$$

(i) Let $f$ be a submersion, $x \in X$ and $y=f(x)$. Show that there exist local coordinates around $x$ and $y$ such that $f$, in these coordinates, is the canonical submersion. [You may assume the inverse function theorem.]

(ii) Show that submersions map open sets to open sets.

(iii) If $X$ is compact and $Y$ connected, show that every submersion is surjective. Are there submersions of compact manifolds into Euclidean spaces $\mathbb{R}^{k}$ with $k \geqslant 1$ ?