TITLE: The Taylor expansion of the metric at the origin in geodesic coordinates
QUESTION [27 upvotes]: It is well known that in geodesic coordinates we have
$$
g_{ij}=\delta_{ij}-\frac{1}{3}\sum_{k,l}R_{ijkl}x^{k}x^{l}+O(|x|^{3})
$$
I have been trying to find a rigorous proof of it, but I cannot find a readable proof online (see this one, for example) or derive it myself. From the compatibility with metric we have
$$
\partial_{i}g_{jk}=\sum_{\alpha}(\Gamma^{\alpha}_{ij}g_{ak}+\Gamma^{\alpha}_{ik}g_{aj})
$$
Since we know that $\nabla_{X}X=0$ for any vector from the origin, we have $\nabla_{i}\partial_{j}=0,\forall i,j$. This then implies all $\Gamma^{k}_{ij}=0$. So the first order derivative vanishes. However, it is not clear to me how to compute the second derivative. We have:
$$
\partial_{l}\partial_{i}g_{jk}=\sum_{\alpha}(\partial_{l}\Gamma^{\alpha}_{ij}g_{\alpha k}+\partial_{l}\Gamma^{\alpha}_{ik}g_{\alpha j})
$$
This can be simplified further by noticing that in the origin we have $g_{ij}=\delta_{ij}$. Therefore the above sum is in fact
$$
\partial_{l}\partial_{i}g_{jk}=\partial_{l}\Gamma^{k}_{ij}+\partial_{l}\Gamma^{j}_{ik}
$$
And we would have proved the statement if we can show that
$$
\frac{1}{2}(\partial_{l}\Gamma^{k}_{ij}+\partial_{l}\Gamma^{j}_{ik})=-\frac{1}{3}(R_{ijkl}+R_{ljki})
$$
By definition we have
$$
R_{abcd}=g_{ae}R^{e}_{bcd}=\sum_{e}g_{ae}R^{e}_{bcd}=R^{a}_{bcd}=\partial_{c}\Gamma^{a}_{db}-\partial_{d}\Gamma^{a}_{bc}
$$
Therefore we have
$$
R_{ijkl}+R_{ljki}=\partial_{k}\Gamma^{i}_{jl}-\partial_{l}\Gamma^{i}_{jk}+\partial_{k}\Gamma^{l}_{ij}-\partial_{i}\Gamma^{l}_{jk}
$$
But I do not know why we would have
$$
\frac{1}{2}(\partial_{l}\Gamma^{k}_{ij}+\partial_{l}\Gamma^{j}_{ik})=-\frac{1}{3}(\partial_{k}\Gamma^{i}_{jl}-\partial_{l}\Gamma^{i}_{jk}+\partial_{k}\Gamma^{l}_{ij}-\partial_{i}\Gamma^{l}_{jk})
$$
I do not know if I missed something obvious like using the Bianchi identity.

REPLY [4 votes]: I asked this many years ago as a beginning graduate student. One general method of solving it is using Jacobi fields, and it can be found for example in Schoen-Yau's book. If I recall correctly, it is listed as an execrise problem in Morgan-Tian's book. The first one proving it is apparently Riemann himself. 
I do not know if my original method has any hope (it has been four years, so maybe not). Hopefully this helps other people.