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?

If your two paths are homotopic you can make a comparison between your two vectors. And yes the comparison involves an integral over the homotopy of a function of curvature. 
See for example Theorem 13.6.4 in Pressley's "Elementary Differential Geometry" (Google books will bring up the statement of the theorem). 
If your paths are not homotopic you're out of luck, as T describes. There are things you can say of course but it's not clear what you're looking for.  You should think of an example of a Riemann manifold you're interested in, to get a sense for how bad it can get. 
