Is there any difference between a flat manifold and an affine space?

Question

What is the difference, if any, between a flat manifold (in which the Riemann tensor vanished identically) and an affine space?

Answer 1

A Riemmanian manifold is called flat if its curvature vanishes everywhere. However, this does not mean that this is is an affine space. It merely means (roughly) that locally it "is like an Euclidean space."

Examples of flat manifolds include circles (1-dim), cyclinders (2-dim), the Möbius strip (2-dim) and various other things.

Yet, let me add into the direction of your idea that the universal cover of a complete flat manifold is indeed an Euclidean space.

Answer 2

If $G < Isom (\mathbb{R}^n) \cong O(n) \rtimes \mathbb{R}^n$ is a subgroup of isometries of some Euclidean space $\mathbb{R}^n$ acting freely and properly discontinuously on it, then the quotient will be a flat Riemannian manifold. Conversely, any flat Riemannian manifold arises in this way. In the realm of surfaces, this gives you planes, disks, annuli(=cylinders), Mobius bands, tori and Klein bottles. As you see you can have non trivial topology (the main constraint in this case is that the Euler characteristic must vanish, by Gauss-Bonnet).

On the other hand, an affine space is always a contractible space: fix any point $P$, then retract the whole space onto $P$ by straight line homotopy, i.e. $(x,t)\mapsto tP + (1-t)x$. Thus the only surface admitting the structure of an affine space is the plane.

Moreover, on a flat manifold you have well defined geo-metric notions: for example it makes sense to measure angles or lengths on a flat manifold (actually any Riemannian manifold). But it makes no sense to measure such entities on an affine space (such measurements do not belong to affine geometry). From this point of view, the structure of an affine space is more algebraic than geometric: here you have a way to add vectors to points or to perform affine combinations (something which is not available on a general Riemannian manifold). So I guess the point here is that the two structures belong to two different realms of "geometry" and are not really comparable to each other.