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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 $\bigl\{ \mathbf{v} \in \mathbf{R}^n : \lVert\mathbf{v}\rVert < 1 \bigr\}$.

For what manifolds does there exist an atlas of charts $c : U \to B_n$ such that the transition maps are all locally affine?

(Replacing "locally affine" with "piecewise affine" would make the answer, by definition, those manifolds for which there exists a piecewise linear structure.)

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Are you looking for sufficient conditions or a list of examples? –  Misha Nov 18 '12 at 13:43
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up vote 10 down vote accepted

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.

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David -- a small remark: the Chern conjecture is for compact affine manifolds; otherwise it is obviously false (take the Euclidean 3-space minus a point). –  algori Nov 18 '12 at 14:23
    
algori-- thank you for your remark. –  David C Nov 18 '12 at 14:44
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