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.)