It is a well known fact that the geometric meaning of a linear connection's curvature can be realized as the measure of a change in a fiber element as it is parallel transported along a closed loop.

As far as I am aware, the formula goes as $$ F(X,Y)\sigma=-\lim_{t,s\rightarrow 0}\frac{1}{ts}\left(P^X_tP^Y_sP^X_{-t}P^Y_{-s}\sigma-\sigma\right), $$ where $P^X_t$ is parallel transport along $X$'s integral curves for time $t$, and I here assume that $[X,Y]=0$.

Unfortunately, none of my references contain a proof of this statement, at least not one that is useful to me now.

I can prove this statement using coordinate-based methods that often involve "expanding to first order" and other rather handwave-y methods (Weinberg, Wald etc. though Wald's procedure is actually fairly interesting and rigorous, it is not what I am looking for), but that's not what I am looking for.

I have also seen proofs of this statement (Lee's Manifolds and Differential Geometry) which involves starting with vectors $u$ and $v$ at points and extending them to vector fields in special ways, so that they are parallel along some curves, which simplifies things greatly. This is once again not what I am looking for, because it already presupposes that we know that $F$ is tensorial and that $F(X,Y)=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]}$.

In essence, what I am looking for is a *derivation* that $$ -\lim_{t,s\rightarrow 0}\frac{1}{ts}\left(P^X_tP^Y_sP^X_{-t}P^Y_{-s}\sigma-\sigma\right)=\nabla_X\nabla_Y\sigma-\nabla_Y\nabla_X\sigma $$ for commuting vector fields $X,Y$, **without supposing we already know what curvature is** (I need this for didactic reasons, for possible use in a general relativity lecture).

I have tried to do this on my own, but I am really terrible at formally manipulating these "flow" type of objects like $P^X_t$, and so far my attempts failed.

I would appreciate any reference, textbooks, lecture notes, articles, papers, which *prove* this formula (with the added caveat that I outlined in bold above). If it does so without assuming $X$ and $Y$ commute, it would especially be stellar, but I don't *need* that.

geodesic(with respect to $\nabla$) along the direction $X\in T_p M$ at time $t$, not the flow of a vector field extending $X$ locally. Or maybe it's the same? $\endgroup$