Let $F: V \times W\rightarrow Z$, where $V,W,Z$ are finite-dimensional smooth (or analytic) manifolds and $F$ is smooth (or analytic). Assume that $\dim W=\dim Z$ and the usual inverse function theorem applies when we fix the first argument of $F$, so locally we have an inverse (w.r.t. the second argument of $F$) mapping $f: V \times Z \rightarrow W$ such that for any $v,w,z$ from suitably chosen small vicinities we have $F(v,f(v,z))\equiv z$ and $f$ is smooth (or analytic). Here comes the question:

Is there any sensible way to glue together these $f$'s so that we can speak of them globally, and under which conditions?

More precisely, is there a way to treat this situation "as if" $f$ were defined globally, without doing the "restrict ourselves to suitably chosen small vicinities" thing all the time, even if we allow the Jacobian of $F$ w.r.t. $w$ to vanish at some points or even on some submanifolds of $V\times W$ of nonzero codimension?

I do not quite expect that (under reasonably mild conditions) one can construct a global $f$ defined on the whole $Q=Z\times V/Sing$ where $Sing=\lbrace(v,z)\in V\times Z|f \quad \mbox{can't be reasonably defined}\rbrace$ but perhaps one can have a sheaf of such $f$'s on $Q$ or something of the sort? I just started with learning the sheaf theory, so I don't quite (yet) know how to make things work on my own.

While in my particular application I would like $V$ to be a manifold, if this makes things easier, I can quite well make do with $W$ and $Z$ being just open domains in $\mathbb{R}^n$ or $\mathbb{C}^n$ (or whole $\mathbb{R}^n$ or $\mathbb{C}^n$ for that matter); that's the main reason why I wrote the domain of $F$ as $V\times W$ rather than just some general manifold $M$.

In fact, I am quite convinced that things like that should have been treated in the literature but I wasn't able to google up anything reasonable so far (maybe I just don't know the right keywords), so pointing out any suitable references is most welcome.

genuinelyglobalize $f$, that's the whole point. $\endgroup$ – anonymous Nov 29 '10 at 5:41