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.
