For all non-negative integers $n$, $B_n$ is defined to be $\: \big\{\mathbf{v} \in \mathbf{R}^n : ||\mathbf{v}||<1\big\} \:$.

For what manifolds does there exist an atlas of charts $\: c : U\to B_n \:$ such that the transition

maps are all locally affine and the union of the domains of the charts is the entire manifold?

(Replacing "locally affine" with "piecewise affine" would make the answer, by definition,

those manifolds for which there exists a piecewise linear structure.)

By the definition I'm using, all manifolds are Hausdorff and second countable.