Transformation groups of certain flat affine manifolds

TL;DR

This paper characterizes the affine transformation groups of certain flat affine manifolds, providing insights into their structure and applications, especially for simply connected cases with diffeomorphic developing maps.

Contribution

It offers a new characterization of affine transformation groups for flat affine simply connected manifolds with diffeomorphic developing maps, using properties of homeomorphisms of Euclidean space.

Findings

Characterization of affine transformation groups for specific flat affine manifolds.

Examples demonstrating the usefulness of the characterization.

Application of homeomorphism properties in the analysis of affine transformations.

Abstract

In this paper we characterize the group of affine transformations of a flat affine simply connected manifold whose developing map is a diffeomorphism. This is proved by making use of some simple facts about homeomorphisms of $\mathbb{R}^n$ preserving open connected sets. We show some examples where the characterization is useful.