# Is every local diffeomorphism from $U \subseteq \mathbb{R}^k$ to a $V \subseteq M$ equivalent to a chart?

If we have a local diffeomorphism $$F: U \rightarrow V$$, with $$U \subseteq \mathbb{R}^k$$ and $$V \subseteq M$$, and $$\mathbb{R}^k$$ with the usual smooth structure, then is it always the case that we can consider $$F^{-1}$$ as a chart map for $$M$$?

It seems so because as a diffeomorphism, $$F \circ \psi^{-1}$$ implies compatibility with the charts of $$M$$.

## 1 Answer

Given any point $$p$$ in $$F(U) \subseteq V$$ and $$x \in F^{-1}(\{p\})$$ there exists a neighbourhood $$W \subseteq U$$ around $$x$$ such that $$F(W)$$ is open with respect to $$U$$ and $$F \vert_W: W \rightarrow F(W)$$ is a diffeomorphism (by definition of local diffeomorphism).

Now we have a problem, since there is no guarantee that $$W$$ is open with respect to $$\mathbb{R}^k$$. Regardless, if $$(X,\phi)$$ is a chart for $$U$$ (seeing $$U$$ as an embedded submanifold of $$\mathbb{R}^k$$) with $$x \in X$$, then $$(F(W \cap X), \phi \circ F\vert_{W \cap X}^{-1})$$ is a chart for $$M$$.

In the special case where $$U$$ is open, $$W$$ is also open in $$\mathbb{R}^k$$, so that $$(F(W), F\vert_W^{-1})$$ is a chart for $$M$$.