The group of affine transformations of homogeneous spaces with discrete isotropy

Abstract

We present a method to compute the group of affine transformations of a homogeneous $G$-space under specific conditions: when the group $G$ and the homogeneous $G$-space admit linear connections so that the natural projection is affine, and with discrete isotropy group. If $G$ admits a bi-invariant linear connection, we establish conditions under which the homogeneous space admits an invariant linear connection. As a consequence, when the isotropy group is discrete, their respective groups of affine transformations are locally isomorphic. As an application of our work, we calculate the group of the affine transformations of orientable flat affine surfaces and 3-dimensional flat affine tori.