Let $E$ be a vector bundle over a smooth manifold $M$ equipped with a linear connection $\nabla : \Gamma(E) \to \Omega^1(M;E).$ I say $(M,E,\nabla)$ is flat if it admits trivial local models; i.e. if for each $p \in M$ there is a $\nabla$-parallel local frame for $E$ defined on some neighbourhood of $p$. It is well known (and often instead taken as the definition) that $(M,E,\nabla)$ is flat if and only if the curvature form $R^\nabla \in \Omega^2(M; E)$ vanishes; so the curvature can be motivated as an obstruction to flatness.

When $E = TM$ so that $\nabla$ is an affine connection, a more restrictive definition is often used: we say $(M,\nabla)$ is flat if each $p \in M$ is contained in a chart whose coordinate frame is $\nabla$-parallel. This imposes an additional requirement on the local model: not only must we be able to choose a frame making $\nabla$ trivial, but this frame must be holonomic. Again, a nice characterization of this kind of flatness is well-known: it's equivalent to both the curvature $R^\nabla$ *and* the torsion $T^\nabla$ vanishing.

Thus in the world of affine connections that are flat in the weaker sense ($R^\nabla = 0$) where we can always find a parallel frame, torsion is exactly what obstructs integrating such a frame to get a chart. This interpretation doesn't feel very satisfying to me, however, since it applies only to flat connections, while in practice we are much more likely to restrict to torsion-free connections before even thinking about curvature.

Question.Without the assumption that $R^\nabla = 0$, can we motivate torsion as the obstruction to some kind of integrability?

This is motivated in part by this nice answer to a broader question, which describes torsion as an obstruction to the integrability of various $G$-structures; but I'm having trouble seeing how this interpretation can apply in the case of a connection alone.

This question received no answer after I posted a bounty on Math.SE.

Pleasedo not ask the same question in both sites. If you ask moderators, questions can bemoved. $\endgroup$ – Mariano Suárez-Álvarez Jan 4 '18 at 3:27