I wonder whether it is always possible to bring a 2D Riemannian metric to a diagonal form with determinant one by changing the coordinates, i.e. for the line element
ds^2 = A(x,y), dx^2 + B(x,y), dy^2
to obtain $A(x,y), B(x,y) = 1$ everywhere.
It is known that one can bring the metric to a diagonal and conformally flat form
ds^2 = C(x,y), (dx^2 + dy^2)
but it is not enough yet.