There's a close relationship between curvature and the holonomy group; the holonomy theorem of Ambrose and Singer, for example. It seems to me that there should be an analogous result for torsion. I believe that torsion measures the extent to which certain geodesic parallelograms don't close. Is there a theorem that makes this more precise?



Torsion and parallel transport
Here is another way to think of the relation between torsion and parallel transport, one that some may find more congenial than many of the other interpretations that have been proposed:


Start with a manifold $M$, and a connection $\nabla$ on $TM$. Consider the bundle $\hat TM=\mathbb{R}\oplus TM$ (where, by $\mathbb{R}$, I really mean the trivial bundle $M\times\mathbb{R}$). Define a connection $\hat\nabla$ on $\hat TM$ by the rule
$$
\hat\nabla\_X\ \begin{pmatrix}a \\\\ Y\end{pmatrix} 
= \begin{pmatrix}da(X) \\\\ aX + \nabla\_XY\end{pmatrix}
$$
for any function $a$ on $M$ and any vector fields $X$ and $Y$ on $M$.


I like to think of $\hat TM$ as a sort of 'extended' tangent bundle to $M$. Of course, it canonically contains $TM= 0\oplus TM\subset \mathbb{R}\oplus TM$ as a subbundle, and one sees that, under this identification of $\begin{pmatrix}0\\ Y\end{pmatrix}$ with $Y$, one has $\hat\nabla\_X Y = \nabla\_XY$. Also, by its definition, $\hat\nabla$-parallel translation in $\hat TM$ along a curve in $M$ will keep the $\mathbb{R}$-component of a section constant, so that the 'affine hyperplane subbundles' $a=const$ in $\hat TM$ are preserved under $\hat\nabla$-parallel translation.


Now, the curvature endomorphism of $\hat\nabla$, namely
$$
R^{\hat\nabla}(X,Y) = \hat\nabla\_X\hat\nabla\_Y-\hat\nabla\_Y\hat\nabla\_X - \hat\nabla\_{[X,Y]},
$$
is just
$$
R^{\hat\nabla}(X,Y)\begin{pmatrix}a \\\\ Z\end{pmatrix}
= \begin{pmatrix}0&0\\\\T^{\nabla}(X,Y) & R^{\nabla}(X,Y)\end{pmatrix}
\begin{pmatrix}a \\\\ Z\end{pmatrix}.
$$


Thus, the condition that $T^\nabla(X,Y)=0$ is the condition that $\hat\nabla$-parallel translation around closed loops preserve the splitting $\hat TM = \mathbb{R}\oplus TM$, at least to lowest (i.e., second) order. In particular, if one thinks of the hyperplane subbundle $a=1$ as a sort of 'shadow copy' of the actual tangent plane, the torsion measures how its 'zero', i.e., $\begin{pmatrix}1\\0\end{pmatrix}$, moves in that 'shadow tangent plane' when one uses $\hat\nabla$ to parallel translate around in a loop.


In particular, note that the curvature of $\hat\nabla$ incorporates both the torsion and curvature of $\nabla$, and the vanishing of the torsion of $\nabla$ says precisely that $\hat\nabla$-parallel translation preserves, to lowest order, the origins in the 'shadow copies' of the tangent plane.


NB: It is a notational accident that, in this interpretation, $T(X,Y)$ represents 'infinitesimal translation' while $R(X,Y)$ represents 'infinitesimal rotation'.


**Remark 1:** The addition of a line bundle to the tangent bundle may seem somewhat artificial. However, there is a way to present this construction more naturally in a dual formulation: First, note that the vector bundle $J^1(M,\mathbb{R})$ of $1$-jets of smooth functions on $M$ has a natural identification with the vector bundle $\mathbb{R}\times T^\*\!M$ by identifying the element $j^1\_xf$ with $(f(x), \mathrm{d}f\_x)\in \mathbb{R}\times T^\*\!M$.
Second, note that a connection $\nabla$ on $TM$ naturally extends to a connection on $T^\*\!M$, uniquely defined by the conditions that
$X\bigl(\eta(Y)\bigr) = (\nabla\_X\eta)(Y) + \eta(\nabla\_XY)$
for all $1$-forms $\eta$ and vector fields $X,Y$ on $M$. Using this identification of $J^1(M,\mathbb{R})$ with $\mathbb{R}\times T^\*\!M$, we can define a connection $\hat\nabla$ on $J^1(M,\mathbb{R})$ by
$$
\hat\nabla\_X(a,\alpha) = \bigl(Xa + \alpha(X),\ \nabla\_X\alpha\bigr)
$$
for all functions $a$, $1$-forms $\alpha$, and vector fields $X$ on $M$.
Then one easily computes that
$$
R^{\hat\nabla}(X,Y)(a,\alpha) = \bigl(\alpha\bigl(T^\nabla(X,Y)\bigr),\ R^{\nabla}(X,Y)(\alpha)\bigr),
$$
which is the dual of the above formula on the 'extended tangent bundle'.


**Remark 2:** Finally, going back to the OP's original request for how to think about torsion in terms of failure of 'quadrilaterals' to close, I checked on the coordinate expressions and looked at the Taylor series. This is, of course, very classical; it's in Schouten's work, but it might be worth making explicit in the following way: Let $\nabla$ be a connection on $TM$, fix a point $p$ and $p$-centered coordinates $x=(x^i)$. Define the connection coefficients $\Gamma^i\_{jk}$ by the usual rule
$$
\nabla\_{\partial\_i}\partial\_j = \Gamma^k\_{ij}\ \partial\_k\ ,
$$
where $\partial\_i$ are the dual vector fields, i.e., $dx^i(\partial\_j) = \delta^i\_j$.


Choose $v,w\in T\_pM$ be tangent vectors and write $v=v^i\ \partial\_i(p)$ and $w=w^i\ \partial\_i(p)$. Let $a(t)=\exp\_p(tv)$ be the $\nabla$-geodesic starting at $p$ with initial velocity $v$, let $b(t)\in T\_{a(t)}M$ be the $\nabla$-parallel translate along $a$ of $w$, and set $c(s,t) = \exp\_{a(t)}(sb(t))$. Then the functions $c^i= x^i(c(s,t))$ have Taylor expansions in $s$ and $t$ of the form
$$
c^i(s,t) = tv^i+sw^i - \tfrac12\Gamma^i\_{jk}(0)\bigl(t^2v^jv^k+2st\ v^jw^k+s^2w^jw^k\bigr) + R^i\_3(s,t).
$$
Thus, if switching $(t,v)$ and $(s,w)$ in this construction is to yield the same result up to second order in $s$ and $t$ for all $v$ and $w$, one must have $\Gamma^i\_{jk}(0) = \Gamma^i\_{kj}(0)$. In particular, if all the 'attempted parallelograms' close to second order at all points, the torsion of $\nabla$ must vanish.


