The group of affine transformations of homogeneous spaces with discrete isotropy

TL;DR

This paper develops a method to compute affine transformation groups of homogeneous spaces with discrete isotropy, focusing on spaces with linear connections and applying the results to flat affine surfaces and tori.

Contribution

It introduces a new approach to determine affine transformation groups for homogeneous spaces with discrete isotropy, especially when linear connections are present.

Findings

Computed affine transformation groups for flat affine surfaces.

Determined affine groups for 3D flat affine tori.

Established conditions for invariant linear connections on homogeneous spaces.

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.