Flat Affine Manifolds And Their Transformations

TL;DR

This paper characterizes flat affine connections on manifolds using affine representations, explores the action of affine transformation groups, and discusses associated Lie algebra structures, with examples illustrating these concepts.

Contribution

It introduces a natural affine representation for flat affine manifolds, analyzes the action of their transformation groups, and studies associative envelopes of Lie algebras related to these groups.

Findings

Affine transformation groups act with open orbits on r^n when their dimension exceeds n.

When the group dimension equals n, the orbits have discrete isotropy.

Existence of associative envelopes for Lie subgroups of affine transformations.

Abstract

We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental form of the bundle of linear frames of the manifold. We show that the group of affine transformations of a real flat affine $n$-dimensional manifold, acts on $\mathbb{R}^n$ leaving an open orbit when its dimension is greater than $n$. Moreover, when the dimension of the group of affine transformations is $n$, this orbit has discrete isotropy. For any given Lie subgroup $H$ of affine transformations of the manifold, we show the existence of an associative envelope of the Lie algebra of $H$, relative to the connection. The case when $M$ is a Lie group and $H$ acts on $G$ by left translations is particularly interesting. We also exhibit some results about flat affine manifolds whose group of affine transformations admits a flat affine bi-invariant structure. The paper is illustrated with several examples.