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 $M$.


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 $M$.


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