By the definition I'm using, all manifolds are Hausdorff and second countable. For all non-negative integers $n$, I define $B\_n$ to be $\bigl\{ \mathbf{v} \in \mathbf{R}^n : \lVert\mathbf{v}\rVert < 1 \bigr\}$.



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> For what manifolds does there exist an atlas of charts $c : U \to B\_n$ such that the transition maps are all locally affine?
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(Replacing "locally affine" with "piecewise affine" would make the answer, by definition, those manifolds for which there exists a [piecewise linear structure](http://en.wikipedia.org/wiki/Piecewise_linear_manifold).)



What manifolds can have a (non-piecewise) linear structure?
If I am right, such manifolds are called affine manifolds. They are smooth manifolds together with a flat, torsion free connection.
Maybe it is worth recalling Chern's conjecture that the Euler characteristic of an affine manifold should vanish.


Kostant B. and Sullivan D. in: 
"The Euler characteristic of an affine space form is zero, Bull. Amer. Math. Soc. 81 (1975)", no. 5, 937-938
proved this conjecture in the case of the quotient of the ordinary space ${\mathbb R}^n$ by a discrete group of affine transformations. 


