Flat torsion-free connections

Question

Can a manifold admit two different flat torsion-free connections? By different, I mean that there is not any diffeomorphism that makes the two equal. The question arises from me trying to explain to myself and to others an intuitive way to see two different flat connections over the same manifolds. The first attempt would be to find two non-isometric flat metrics over the same manifold that induce two different connections, but it looks to me like it is not possible.

Answer 1

The Levi-Civita connection on a flat torus is flat and torsion-free as was remarked in the comments. There are other ways to define torsion-free and flat connections. Such a connection is equivalent to an affine structure, that is, an atlas whose charts are related by affine transformations.

Consider an affine torus defined by taking the quotient of the plane minus the origin $ \mathbb{R}^2 \backslash \lbrace 0 \rbrace$ by the group of dilations generated by $ x \mapsto 2x $. The natural linear connection on this is not equivalent to one given by a flat metric. For instance, the constructed affine torus is not geodesically complete.

I'm pretty certain that you could find two distinct connections that are both complete but can't think of an easy one right now.