This is, in a sense, a follow up to this question.

Hehl and Obukhov try to give an intuitive description of torsion. I am confused about their description. I am looking at the following paragraph on page 5:

How can a local observer at a point $p$ with coordinates $x^i$ tell whether his or her space carries torsion and/or curvature? The local observer defines a small loop (or a circuit) originating from $p$ and leading back to $p$. Then he/she rolls the local reference space without sliding ... As a computation shows, the added up translation is a measure for the torsion and the rotation for the curvature.

So, my input data is a manifold $M$ with a connection $\nabla$ on $T_* M$, and a path $\gamma : [0,1] \to M$. From this data, I am supposed to obtain an affine linear map $v \mapsto Av+b$ from $T_{\gamma(0)} M$ to $T_{\gamma(1)}(M)$. In particular, if $\gamma(0)=\gamma(1)=p$, I obtain an endomorphism of $T_p M$. Then $b$ is a measure of the torsion, and $A$ is a measure of the curvature.

I am happy with the curvature part. This is parallel transport: Given a tangent vector $u \in T_p M$, I am to find the unique local section $\sigma$ of $\gamma^* T_p(M)$ such that $\sigma(0) = u$ and $\nabla \sigma=0$. Then $Au = \sigma(1)$.

I have two confusions about the torsion part:

(1) Hehl and Obukhov cite their definition of rolling without slipping to five sources (hidden by the ellipses above). The most readable of the ones I have access to is *Differential Geometry*, by Sharpe. But Sharpe (at least in Appendix B) only gives a definition for the Levi-Civita connection, not for an arbitrary connection. I think I have guessed what the definition should be, but could someone please write it down so I can be sure?

(2) In any case, it is my understanding that the Levi-Civita connection should be torsion-free and that, for the Levi-Civita connection, rolling without slipping corresponds to physically rolling my manifold along a plane. So the statement should be, if I take a surface $S$ in $\mathbb{R}^3$, and physically roll it along a table top, when I get back to the same tangency point on $S$, I should be at the same point on the table. Physical experimentation has not made it clear to me whether or not this is true. However, if I roll my surface along a non-small path, this is definitely false: otherwise, ball bearings could not roll!

Technically, this is not a contradiction, because Hehl and Obukhov only speak of a small circuit. But the usual situation in differential geometry is that when some quantity vanishes everywhere on a manifold, then "small" can be replaced by "contractible". And the example of a rolling ball bearing definitely shows that rolling-without-slipping a plane along a contractible loop according to the LC connection can produce a nontrivial translation.

What's going on?