Second derivative of the Christoffel symbols in normal coordinates

Question

According to wikipedia the Taylor expansion of the Christoffel symbols of a Riemannian manifold $(M,g)$ in normal coordinates is given by $$ {\Gamma^{\lambda}}_{\mu\nu}(x)= -\frac 13 (R_{\lambda\nu\mu\tau}(0)+R_{\lambda\mu\nu\tau}(0))x^\tau+ O(|x|^2). $$ Is there any reference for the calculation of the $O(|x|^2)$ term? I did the computations as Yuval proposed and ended up with $$ 6\dfrac{\partial }{\partial x_v} \dfrac{\partial }{\partial x_w}{\Gamma^{n}}_{ij}(x) \\= \frac 12 g^{nl} \left( \nabla_i R_{jvwl} + \nabla_i R_{jwvl} +\nabla_w R_{jivl} + \nabla_v R_{jiwl} + \nabla_w R_{jvil} + \nabla_v R_{jwil} \right) + \left( \nabla_j R_{ivwl} + \nabla_j R_{iwvl} +\nabla_w R_{ijvl} + \nabla_v R_{ijwl} + \nabla_w R_{ivjl} + \nabla_v R_{iwjl} \right) - \left( \nabla_l R_{ivwj} + \nabla_l R_{iwvj} +\nabla_w R_{ilvj} + \nabla_v R_{ilwj} + \nabla_w R_{ivlj} + \nabla_v R_{iwlj} \right) $$ If I apply the symmetries of the curvature tensor $$ \left( \nabla_i R_{jvwl} + \nabla_i R_{jwvl} + 2\nabla_w R_{jvil} + 2\nabla_v R_{jwil} \right) + \left( \nabla_j R_{ivwl} + \nabla_j R_{iwvl} + 2\nabla_w R_{ivjl} + 2\nabla_v R_{iwjl} \right) - \left( \nabla_l R_{ivwj} + \nabla_l R_{iwvj} \right) $$ I feel like the derivatives involving derivatives in $i,j,l$ should somehow cancel out with the Bianchi identities but I don't get it to work.

Answer 1

This needs some computations involving the Jacobi field ODE. You just need an Taylor expansion for $g_{ij}$ since $\Gamma_{ij}^k$ involves $g$ and its first derivative. See

John Lee & Thomas Parker "Yamabe Problem", Bull. AMS. vol 17. no 1 (1987), see page 60-61, especially equation (54).

or

Schoen & Yau "Lectures on Differential Geometry", p210-211. You can use (3.4) there to compute the $O(|x|^2)$ term.