Is the deformation space of complete affine structures on the 2-torus smooth?

Oliver Baues, William M. Goldman

Published 2004-01-20, updated 2005-06-14Version 2

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 $R^2$.