  Periods of parallel exterior forms define natural coordinates on the
deformation space of complete affine structures on the two-torus. These
coordinates define a differentiable structure on this deformation space, under
which it is diffeomorphic to $R^2$. The action of the mapping class group of
$T^2$ is equivalent in these coordinates with the standard linear action of
$\SL_2(Z)$ on $R^2$.
