I am trying to get a geometric interpretation of the riemann curvature tensor for 2-dimensional surfaces: $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ In https://en.m.wikipedia.org/wiki/Riemann_curvature_tensor we can find the following equality: $$\frac{d}{ds}\frac{d}{dt} \tau_{sX} ^{-1} \tau_{tY}^{-1}\tau_{sX} \tau_{sY} \Big|_{s=t=0}=R(X,Y)Z$$ Where I can get the proof of the last statement?
Many thanks!
The equation you are stating is meaning, that if you take a vector $v$ at some $p\in M$ and you parallel transport it along a rectangle on a two-dimensional surface in $M$ the resulting deviation for the limit of a vanishing rectangle is exactly what one defines as "curvature". For an explicit calculation leading to this interpretation of Curvature I refer you to read the second and third chapter of
R. Wald, General Relativity, Chicago University Press, 1984