Affine structures I would like to study manifolds endowed with a linear connexion $\nabla$ which is torsion free and locally flat i.e. its curvature is $0$ (such a connexion is called flat if in addition, its holonomy is trivial).
I am looking for a good reference:


*

*to prove that such a manifold admits an atlas with affine transition maps (restriction of such maps on opens of $\mathbb{R}^n$).

*to characterize the universal covering and the fundamental group of such manifolds and its link to Bieberbach groups.
Is it true that a compact affine manifold has a trivial Euler characteristic? (Chern conjecture)
Thank you for any help!
 A: Kostant and Sullivan proved that the Euler characteristic of a compact complete affine manifolds must vanish, affirming the Chern conjecture in the complete case (Bull. AMS 81 (1975)). Benzecri proved that a closed surface which admits a (possibly imcomplete) affine structure must have zero Euler characteristic.
In general a good reference is the book "Spaces of constant Curvature" by Joseph A. Wolf (in particular for the link to Bieberbach groups). Also, several survey articles by W. Goldman on affine manifolds are a very good reference, e.g. Locally homogeneous manifolds, see page $11$ for recent work on Chern's conjecture.
A: Here is a sketch of why you get a $(G,X)$ structure.  On any sufficiently small open set you can find a parallel frame for the connection, which by the torsion free hypothesis consists of mutually commuting vector fields.  Thus, these can be simultaneously integrated to give you coordinates.  On competing overlaps, the frames are related by linear maps, and since the connection is flat, these linear maps are locally constant.  Since the transition maps between these frames are linear maps, the transition maps between the coordinates are given by affine maps.  
A: Here is a link in which I propose a possible solution:
https://arxiv.org/pdf/1504.04852.pdf
Any comments are highly appreciated.
