Diffeomorphic real-analytic maps and the Jacobian Conjecture

It is shown that if the real-analytic map f(x) : R2→R2 has a Jacobian matrix whose eigenvalues are always both one, then the map is a diffeomorphism. An explicit form of the inverse is given. The proof relies on a result which says that the only global solutions to the quasi-linear partial differential equation (cos u)ux − (sin u)uy = 0 are constant functions. The main result is put into a context concerning the Jacobian Conjecture.