Parallel transport of a vector along two distinct curves Let $\mathcal{M}$ be an n-dimensional manifold endowed with an affine connection $\nabla$. Let $\gamma_1:[a,b]\rightarrow M$ and $\gamma_2:[c,d]\rightarrow \mathcal{M}$ be two curves with the same initial and final points, that is,
$p=\gamma_1(a)=\gamma_2(c), q=\gamma_1(b)=\gamma_2(d)$. Take  $X\in T_p\mathcal{M}$. Parallelly propagating $X$ along $\gamma_1$ and $\gamma_2$ we obtain two vectors $X_1, X_2\in T_q\mathcal{M}$, respectively. Let $R$ be the curvature tensor of the connection, $R(X,Y)Z=\nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z -\nabla_{[X,Y]}Z$, and $\tau$ its torsion, $\tau(X,Y)=\nabla_X Y-\nabla_Y X -[X,Y]$. 
The question is: How can I compare the two vectors $X_1$ and $X_2$? Can I write the difference $(X_2-X_1)$ in terms of $R,\tau$ and the curves?

The difference depends on the homotopy classes of the paths, not only local data like the curvature and torsion.  I don't think there is a simpler answer than "the difference in the parallel transport along the two paths".  But see the (superb) Wikipedia page on holonomy, especially the sections on the Ambrose-Singer theorem and affine connections.  
http://en.wikipedia.org/wiki/Holonomy
