in my research, I need to understand the relation between the formal definition of Riemann curvature tensor ($R_{jkl}^i$)and the ``vector rotation" approach: parallel transport a vector along a loop, then define the curvature tensor by $$ \lim_{\epsilon\rightarrow 0}\frac{{\hat{\xi^k}(\epsilon)}-\xi^k}{\epsilon^2}=-R_{lij}^k \xi^l $$ where $\hat{\xi}$ is the vector after rotation, along a square with edge length $\epsilon$.
(This formula is given in an exercise of Novikov et al. book "Modern Geometry")
I referred to Wiki, but not quite understand yet...
Is there any detailed material on such issue, or how to establish such relation?
Thanks a lot!