I would like to prove that the existence of an affine atlas (smooth atlas with all transition maps are affine) on a given smooth manifold $M$ is equivalent to the existence of a flat and torsion free connection on $M$.

I tried to the following: If $M$ posses an affine atlas than we can pull back the flat Levi-Civita connection of $\mathbb{R}^n$ on open charts and use a subordinate partition of unity to get an affine connection on $M$. I don't know how to proceed reciprocally.

Although it's a well known result, I don't know where to find a clear treatment of this question.I thank everyone who can give me some references and some hints to solve this problem.


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