intuition - What is torsion in differential geometry intuitively?

Torsion is easy to understand but this knowledge seems to be lost. I had to go back to Elie Cartan's articles to find an intuitive explanation (for example, chapter 2 of http://www.numdam.org/numdam-bin/fitem?id=ASENS_1923_3_40__325_0).

Let $M$ be a manifold with a connection on its tangent bundle.
The basic idea is that any path $gamma$ in $M$ starting at $xin M$ can be lifted as a path $tildegamma$ in $T_xM$, but is the $gamma$ is a loop $tilde gamma$ need not be a loop. The resulting translation of the end point is the torsion (or its macroscopic version).

The situation is easy in a Lie group $G$ (which I imagine Cartan had in mind).
$G$ has a canonical flat connection for which the parallel vectors fields are left invariant vectors fields. For this connection the parallel transport is simply the left translation. The Maurer-Cartan form $alpha$ is then the parallel transport to the tangent space $T_1G$ at the identity $1in G$.

If $gamma:[0,1]to G$ is a path in $G$ starting at $1$. $gamma'$ is a path in $TG$ and $alpha(gamma')$ is a path in $T_1M$. $alpha(gamma')$ can be integrated to another path $tilde gamma$ in $T_1M$. Let $gamma_{leq x}$ be the path $gamma:[0,x]to G$, then we define

$$tilde gamma(x) = int_0^xalpha(gamma'(t))dt = int_{gamma_{leq x}}alpha.$$
In the sense given by the connection, $gamma$ and $tildegamma$ have the same speed and the same starting point, so they are the same path (but in different spaces).

If $gamma$ is a loop and $D$ a disk bounding $gamma$,
$tildegamma$ is a loop iff $tildegamma(1)=0in T_1G$.
We have

$$tildegamma(1) = int_gammaalpha = int_Ddalpha.$$
$tildegamma$ is a loop iff this integral is zero.

Now, $alpha$ can be viewed as the solder form for $TG$, so the torsion is the covariant differential $T=d^nablaalpha$. As the connection is flat $T$ reduces to $T=dalpha$.
The Maurer-Cartan equation gives an explicit formula: $T=dalpha = -frac{1}{2}[alpha,alpha]$.
The previous integral is then the integral of the torsion

$$tildegamma(1) = int_Ddalpha = -frac{1}{2}int_D[alpha,alpha]$$
and may not be zero.

The situation is the same for a general manifold, but the parallel transport is not explicit and formulas are harder.

The notion behing this is that of affine connection. As I understand it, an affine connection is a data that authorize to picture the geometry of $M$ inside the tangent space $T_xM$ of some point $x$. If I move away from $x$ in $M$, there will be a corresponding movement away from the origin in $T_xM$ (this is the above lifting of path). If I transport in parallel a frame with me, the frame will move in $T_xM$. Globally the movement of my point and frame is encoded by a family of affine transformations in $T_xM$.

Of course this picture of the geometry of $M$ in $T_xM$ is not faithful.
Because of the torsion, if I have two paths in $G$ starting at $x$ and ending at the same point, they may not end at the same point in $T_xM$.
Because of curvature, even if my two lifts end at the same point, my two frames may not be parallel.
The picture is faithful if $M$ is an affine space iff both torsion and curvature vanish (Cartan's structural equations for affine space).

I think torsion is beautiful :)

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