For $F:R^{n} \to R^{m}, m>n$, with local inverse $G$ at $x$ such that $G(F(x)) = x$, is $DG$ a left inverse of $DF$? Why/Why not?

Question

Assuming both functions are differentiable in some neighboorhood, under which conditions, if any, is the Jacobian of $G$ the left inverse of the Jacobian of $F$, ie $(JG)(JF) = Id^{nxn}$?

Since we're going up in dimension, and then down again, it seems to me as though this should be true for subsets of the domain where $F$ is injective, since the image of $F$ isn't truly of dimension $m$, but an $n-$dimensional space embedded in an $m$ dimensional space. Is this true, and why/why not? What could I read up on to gain more of an understanding? I can find heaps on literature on the $m=n$ case, but I might be searching in the wrong places.

Answer 1

If $G(F(x)) = x$ for $x$ in an open set $U \subset \mathbb{R}^n$, then by the chain rule, $$DG(F(x))DF(x) = I.$$ So $DG(F(x))$ is a left inverse for $DF(x)$.