Curvature and Parallel Transport

It appears to me that one reason why nobody has proved the formula yet is that the formula is still wrong. First, the formula has to depend on $X$ and $Y$. If you rescale $X$ and $Y$, the left side of the formula scales but the right side stays constant. That can't be. Second, the two sides of the equation do not scale the same under a constant scaling of the metric.

Warning: I wrote this up very quickly and did not check for typos and errors. It's possible that my final formula is still not right, but I am confident that my argument can be used to obtain a correct formula. I also did not provide every last detail, so, if you're unfamiliar with an argument like this, you need to do a lot of work making sure that everything really works. The key trick is pulling everything back to the unit square, where elementary calculus can be used. I'm sure this trick can be replaced by Stokes' theorem on the manifold itself, but that's too sophisticated for my taste.

Holonomy calculation

ADDED:

The correct formula, if you assume $|X\wedge Y| = 1$, is

$P_\gamma Z - Z = Area(c) R(X,Y)Z$

This scales properly when you rescale the metric by a constant factor. Notice that the left side is invariant under rescaling of the metric.

I recommend looking at papers written by Hermann Karcher, especially the one with Jost on almost linear functions, the one with Heintze on a generalized comparison theorem, and the one on the Riemannian center of mass. I haven't looked at this or anything else in a long time, but I have the impression that I learned a lot about how to work with Jacobi fields and Riemann curvature from these papers.

Finally, don't worry about citing anything I've said or wrote. Just write up your own proof of whatever you need. If it happens to look very similar to what I wrote, that's OK. I consider all of this "standard stuff" that any good Riemannian geometer knows, even if they would say it differently from me.

EVEN MORE: There are similar calculations in my paper with Penny Smith: P. D. Smith and Deane Yang Removing Point Singularities of Riemannian Manifolds, TAMS (333) 203-219, especially in section 7 titled "Radially parallel vector fields". In section 5, we attribute our approach to H. Karcher and cite specific references.

These matters are discussed in great detail in lecture 19 of the book "Differential geometry" by Postnikov. The original Russian edition is reviewed in [MR0985587 (90h:53002)]; there is also French edition, but I am not aware of an English one.

Your formula for $R(X,Y)Z$ seems to be identical to the formula on page 256 of Peter Petersen, "Riemannian Geometry", second edition, Springer 2006. Petersen gives a sketch of a proof, and calls the formula "Cartan's characterization of the curvature".