The deformation of flat affine structures on the two-torus

Abstract

The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group Aff(2) on ${\mathbb{R}}^2$. Since this action has non-compact stabiliser GL(2,$\mathbb{R}$), the underlying locally homogeneous geometry is highly non-Riemannian. In this chapter, we describe the deformation space of all flat affine structures on the two-torus. In this context interesting phenomena arise in the topology of the deformation space, which, for example, is not a Hausdorff space. This contrasts with the case of constant curvature metrics, or conformal structures on surfaces, which are encountered in classical Teichmüller theory. As our main result on the space of deformations of flat affine structures on the two-torus we prove that the holonomy map from the deformation space to the variety of conjugacy classes of homomorphisms from the fundamental group of the two-torus to the affine group is a local homeomorphism.