TITLE: Rolling without slipping interpretation of torsion
QUESTION [26 upvotes]: 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?

REPLY [2 votes]: This was meant to be a comment over at Secret Blogging but was probably classified as spam (not that I disagree with that), so I stick it here instead.
An example to consider is $S^3$. The metric from the ordinary embedding into $E^3$ gives a constant curvature connection with no torsion. The geodesics are all great circles and splits into equivalent classes of parallel geodesics (the Hopf fibration). An observer travelling along a geodesic will observe how nearby geodesics twist around him. This is a higher dimensional analogue of how nearby geodesics in two dimensions are observed to do sinusidal oscillations when the curvature is positive.
The curvature form is an so(3)-valued two-form which integrated around (the interior of) a closed loop gives the rotation of a frame transported around the loop. Now, given an element of so(3) at a point of the manifold, it can be reinterpreted as an vector in the tangent space. (This is the usual so(3) <-> angular velocity vector isomorphism.) This turns the curvature form into a torsion form, giving a new connection with no curvature but with "constant" (homogeneous) torsion. Integrating the torsion form gives a tangent vector which is the translation of the tangent space when translated around a (not necessarily small) loop.
This absolute parallelism connection has the same geodesics as the constant curvature connection. But in this case, an observer travelling along a geodesic, I believe, does not observe any twisting of nearby geodesics. Instead he sees the nearby geodesics to be completely straight lines, but they lag behind (or run ahead?). In some way a torsion connection introduces an ambiguity in the velocity concept which would be interpreted by an observer that the immediate neighbourhood does not stay in place, it slips?
Now, bundles. Take the tangent bundle $TM$ of some manifold and release each tangent space from its point of contact with the manifold. This turns tangent spaces into affine spaces and the tangent bundle into an affine bundle $AM$. This bundle does not have a distinguished zero section, instead each and every point of an affine fiber have equally right to be considered a point of the underlying manifold. Then, giving a connection on $M$, it should the case that the connection has vanishing torsion if and only if every contractible closed loop in $M$ lifts to a closed loop in $AM$?
The connection in $R^3$ with straight lines as geodesics and when a frame is transported along a line, it spins, is given by $\nabla_XY=\nabla_YX=Z$, etc. This is again a constant curvature connection with no torsion.