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.

中文翻译：

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平面仿射流形及其变换

我们通过保持连接的微分同胚李群的普遍覆盖的自然仿射表示，给出了流形上平面仿射连接的表征。从无穷小的角度来看，这种表示是由流形的线性框架丛的1-连接形式和基本形式决定的。我们证明了真正的平面仿射 $n$ 维流形的仿射变换群作用于 $\mathbb{R}^n$，当其维度大于 $n$ 时离开开放轨道。此外，当仿射变换群的维数为$n$时，该轨道具有离散的各向同性。对于流形的仿射变换的任何给定李子群 $H$，我们证明 $H$ 的李代数的关联包络的存在，相对于连接。$M$ 是李群，$H$ 通过左平移作用于$G$ 的情况特别有趣。我们还展示了一些关于平面仿射流形的结果，其仿射变换组允许平面仿射双不变结构。本文通过几个例子进行了说明。