I am learning about the Legendre transform and found in Mac Lane's Geometrical Mechanics lecture notes (v1 p.54) the following inversion theorem:
The Legendre transformation $\ v\in V\mapsto dL(v)\in V^*$ is invertible if and only if the Hessian matrix of $L$ is non singular at each point
where $L:V\mapsto\mathbb{R}$ is a (Lagrangian) function (at a given point in space, V represents the tangent space). Looking further in the literature, there seem to be a distinction made between regular Lagrangians (Legendre transformation is locally invertible) and hyperregular (Legendre transformation is globally invertible). I assume the lecture notes overlooked this distinction.
Looking for a global inversion theorem, I found a paper by W.B. Gordon in which he asserts that
A $\mathcal{C}^1$ map from $\mathbb{R}^n$ to $\mathbb{R}^n$ is a diffeomorphism if and only if $f$ is proper and its Jacobian determinant never vanishes.
I am interested in the second proof, more topological. The paper first asserts that $f$ is onto then concludes that $f$ is a covering of a simply connected space, hence a global diffeomorphism.
It is not proved that $f$ is onto, which seems to use some basic degree theory. I am not very familiar with the degree of maps but I assume we want to prove that the degree is $\pm 1$ so that each point is in the image, according to the formula $\forall y\in\mathbb{R}^n, \operatorname{deg}f = \sum_{f(x)=y} \operatorname{deg}_x f$.
How can I prove that f is onto? Actually I'm not even sure how the degree of the map is defined (do we consider the continuous map between the one-point compactifications? Or maybe we consider the non-compactly supported de Rham top cohomology ?)
Assuming the theorem, it would then seem that a regular Lagrangian is hyperregular provided its associated Legendre transformation is proper.