This is, in a sense, a follow up to [this question](https://mathoverflow.net/questions/20493/what-is-torsion-in-differential-geometry-intuitively).


[Hehl and Obukhov](http://arxiv.org/abs/0711.1535) 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*](http://books.google.com/books?id=poV6SSopE9QC&lpg=PP1&dq=sharpe%2520differential&pg=PP1#v=onepage&q&f=false), 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?



Rolling without slipping interpretation of torsion
"Rolling without slipping" is a powerful idea, but the phrase doesn't necessarily lead one to the intended mental model. In particular, 
torsion is something that is at issue only for manifolds of dimension 3 or higher. Perhaps you can imagine taking a 3-manifold, and rolling it along a hyperplane in 4-space --- but
the metaphor becomes strained, partly because most Riemannian 3-manifolds cannot be 
smoothly isometrically embedded in $\mathbb E^4$.


Another way to think of it is this: suppose you have a smooth parametrized
curve say in a Riemannian
3-manifold, $\alpha: [0,T] \rightarrow M^3$. Then the claim is that there exists
exists a matching curve $\beta: [0,T] \rightarrow \mathbb E^3$ together with a map
$\phi$ from a neighborhood of the image of $\beta$ to a neighborhood of the image of $\alpha$
that takes the Euclidean metric to the metric of $M^3$ up to first order along the curve.
Furthermore, $\beta$ is uniquely determined up to an isometry of $\mathbb E^3$.
Basically, $\beta$ is what you get if you "roll" $M^3$ along $\mathbb E^3$ along $\gamma$
in a way
that best maintains contact between the two spaces: that is, "without slipping".


To see that the curve $\beta$ (assuming it exists) is uniquely determined by $\alpha$,
we can imagine trying to send a neighborhood of $\beta$ to a neighborhood of another
curve $\gamma: [0,T] \rightarrow \mathbb E^3$ in a way that maintains first order
contact of the metric. If you look at curves parallel to $\beta$ in $\mathbb E^3$, the first
derivative of their arc length
is negative in the direction that $\beta$ is curving. The logarithmic derivative of
arc length, for curves displaced along a normal vector field that remains as parallel to
itself as possible, is the magnitude of the curvature.


If you try to twist the normal coordinate system, this corresponds to the concept
of torsion. It's easiest to visualize along a straight line: if you twist a neighborhood
of a line in space,
you distort the metric on each concentric cylinder, by changing the angles between
cross-section circles and generating lines. *I.e.*, threads that wind around a hose
at angles $\pm \pi/4$ are effective at preventing twisting (= torsion). The same
principle holds for any curve in space: the first order behavior of the metric in a 
neighborhood of the curve locks in the Frenet characterization of the curve (well,
the curvature and torsion as a function of arc length, but these
are different from but related to the curvature and torsion of a connection).


Why does the matching curve exist? You can check derviatives etc. but
better to just imagine it. Basically, you could reparametrize $\alpha$ by arc length,
then project a neighborhood of $\alpha$ back to $\alpha$ by sending each point to the closest
point of $\alpha$, and parametrize the lines of projection by their arc length.
On each concentric tube, there's a unique unit vector field orthogonal to the preimages
of projection to $\alpha$. Scale this vector field so that it commutes with
projection, to get a full set of
cylindrical coordinates for a neighborhood of $\alpha$. The only first-order
invariant for the metric that is free is the first derivative of scaling function.
Using that, you can match the first derivative
by using the curvature and torsion of a curve in space.


This process defines the affine connection on the tangent bundle. The Levi-Civita
connection is the linear part of the affine connection, which is automatically by
definition torsion free. The non-torsion-free connections are ones that
impart twists on little neighborhoods of curves. This is usually expressed
by translating it into a formula about covariant derivatives of two vector fields
not being as commutative as it should be.


This really calls for pictures. Any volunteers?


