extension/"globalization" of inverse function theorem I am curious as to what changes do we need to make to the hypotheses of the
inverse function theorem in order to be able to find the global differentiable inverse to a differentiable function. We obviously need $f$ to be a bijection, and $f'$ to be non-zero. Is this sufficient for the existence of a global differentiable inverse?
For functions $f\colon\mathbb{R}\to\mathbb{R}$, we have
Motivation:
$f^{-1}(f(x))=x$, so $(f')^{-1}(f(x))f'(x)=1$
Then, we could define $(f')^{^-1}(f(x))$ to be $1/f'(x)$   ( this is the special case of the formula for the differentiable inverse -- when it exists -- in the IFT)
(and we are assumming $f'(x)\neq 0$) 
In the case of $\mathbb{R}^2$, I guess we could think of all the branches of $\log z$ and $\exp z$, and we do have at least a branch-wise global inverse , i.e., if/when $\exp z$ is 1-1 (and it is , of course onto $\mathbb{C}-{0}$), then we have a differentiable inverse.
I guess my question would be: once the conditions of the IFT are satisfied:
in how big of a neighborhood of $x$ can we define this local diffeomorphism,
and, in which case would this neighborhood be the entire domain of definition of $f$?
I guess the case for manifolds would be a generalization of the case of
$\mathbb{R}^n$, but it seems like we would need for the manifolds to have a single chart.
So, are the conditions of f being a bijective, differentiable map sufficient
for the existence of a global differentiable inverse? And, if $f$ is differentiable, but not bijective,  does the IFT hold in the largest subset of the domain of definition of $f$ where $f$ is a bijection?
Thanks.

There is a theorem ("Introduction to Smooth Manifolds," Lee, Thm 7.15) for differentiable manifolds which says that:

If $F: M \to N$ is a differentiable bijective map of constant rank, then $F$ is a diffeomorphism -- so in particular $F^{-1}$ is differentiable.

Here, the rank of differentiable map $F\colon M \to N$ at a point $p \in M$ is defined to be the rank of its pushforward $F_*\colon T_pM \to T_{F(p)}N$.  (Some authors use the word "differential" for "pushforward," and use the notation $d_pF$ for $F_*$.)
