To guarantee the bijectivity of a map, a necessary condition is the positivity of the Jacobian of the map. In general, we have the following lemma:
A continuously differential map $\mathbf G$ is locally bijective provided its Jacobian, denoted as $\det(J_{\mathbf G})$, does not vanish on the parametric domain, and the global bijectivity of $\mathbf G$ is guaranteed if it is locally bijective on the parametric domain, and the computational domain is simply connected and the restriction of $\mathbf G$ on the domain boundary is bijective.
In our situation, we assume that the computational domain $\Omega$ is simply connected, and a bijective boundary correspondence of the parametric domain $\hat{\Omega}$ and computational domain $\Omega$ is established. Thus in this case, a locally bijective parameterization is also globally bijective.
My questions is: If the Jacobian $\det(J_{\mathbf G})$ is positive everywhere interior the parametric domain $\hat{\Omega}$, is the map $\mathbf G$ bijective? In other words, what is the Jacobian of a bijective map on the domain boundary, positive or zero?