Question

By the definition I'm using, all manifolds are Hausdorff and second countable.
For all non-negative integers $n$, I define $B_n$ is 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?

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

Answer 1

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.

Konstant 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.