Transformation groups of certain flat affine manifolds

In this talk we will present a method to compute the group of affine transformations of the base manifold of certain affine galoisian coverings whose automorphims are affine transformations. Then we will give necessary conditions so that a homogeneous G-space admits an invariant linear connection induced from a left invariant linear connection on G (the reductive case is treated separately). Moreover, we will show that if the connection is bi- invariant the natural projection is an affine map. Furthermore, we will prove that, given an affine homogeneous space G/H, affine transformations of G commuting with the G-action determine affine transformations of G/H. The reciprocal is true when H is discrete. As an application, we exhibit the group of affine transformations of the orientable flat affine surfaces.

Keywords: Flat affine manifolds, Affine transformations, Galoisian coverings, Homogeneous spaces, Reductive homogeneous spaces, Invariant connections, Flat affine surfaces