So, my question is this:
given an affine manifold $X$, and a quotient manifold $Y$ of $X$, is $Y$ necessarily an affine manifold?
If it helps, I'm especially interested in the case that $X$ is not any affine manifold but the affine space of dimension $n$ and I also know that $Y$ is orientable, compact, complete and flat.


edit: By affine manifold I mean a manifold that has an atlas such that the transition maps of the charts are locally restrictions of affine maps. Furthermore I ment the *real* affine space. And lastly, are there any conditions under which $Y$ admits an affine atlas?


edit2: As pointed out in the comments, the answer to my original question was trivially anwserable, therefor I modify my question. Given an quotient manifold $Y$ of an affine manifold $X$. Under which further assumptions is $Y$ an affine manifold? Is especially flattness sufficent, and if so, can someone tell my where I can find a proof for this?


edit3: Okay, I try one more time - this time as specific as I can. I have a group $\Gamma < Aff(\mathbb{E})$, $\mathbb{E}$ being the affine real space of dimension $n$, and $\Gamma$ acts on $\mathbb{E}$ in the natural way, freely and without accumulation points. Now I want to show that $\Gamma$ is an affine crystallographic group and I try to do this by showing that $\Gamma \backslash \mathbb{E}$ is a compact, complete manifold with an affine structure and flat affine connection. So far, all I could show, was that the quotient $\Gamma \backslash \mathbb{E}$ is a compact, orientable manifold and I don't know how to proceed, while apparently for Auslander\* this yields already the claim ($\Gamma$ is affine crystallographic). So i was wondering, which properties, like being orientable for example, give the manifold all the properties I want it to have.
I hope I made myself clear this time, as to what I want.


* "Examples of Locally Affine Spaces", *The Annals of Mathematics*, Second Series, Vol. 64, No. 2, 255-259 (1965).



when is a quotient manifold of an affine manifold affine?
With the 3rd edit, your question become a triviality: The affine space $A^n$ has canonical flat (i.e., zero curvature, zero torsion) linear connection $\nabla$: $\nabla\_{X\_i} X\_j=0$ for all coordinate vector fields. This is a calculus exercise to check that $\nabla$ is invariant under all affine transformations. Therefore, $\nabla$ will descend to a flat affine connection on every quotient $M=A^n/\Gamma$, where $\Gamma$ is a discrete torsion-free group of affine transformations. Geodesics in $(A^n, \nabla)$ are points and Euclidean straight lines with linear parameterization (another calculus exercise), therefore $(A^n, \nabla)$ is geodesically complete. Geodesics in the quotient manifold $M$ are exactly the projections of geodesics from $(A^n, \nabla)$. Therefore, $M$ is also geodesically complete. Compactness or/and orientability are irrelevant here. 


