Suppose I have a smooth manifold with a tangent bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a finite-dimensional smooth manifold.)


I know that in general curvature-free connections do not exist, and that in general torsion-free connections do. But is the existence of a curvature-free connection sufficient to prove the existence of a torsion-free connection?



Can a manifold have a curvature-free connection that is not torsion-free?
Many manifolds have curvature-free (i.e., flat) connections on their tangent bundles. For example, any orientable $3$-manifold $M$ is parallelizable, i.e., its tangent bundle is trivial, so it carries a flat connection (in fact, many flat connections). However, nearly all of these connections will have torsion. In fact, they all will unless $M$ is very special; essentially, $M$ has to be what is known as a flat affine manifold.


I don't understand your last question. For example, the $3$-sphere has a flat connection (just regard it as a Lie group and take the connection that makes the left-invariant vector fields parallel). However, the $3$-sphere certainly does not admit a torsion-free flat connection, since it is compact and simply-connected.


