Christoffel symbols in Riemann normal coordinates I'm studying General Relativity using Misner, Thorne and Wheeler textbook called Gravitation. And I'm stuck on an exercise after chapter 11.6 which is about Riemann normal coordinates. Exercise is the following, one must show that in Riemann normal coordinates at point $\mathcal{P}_0$, for the Christoffel symbols we have the following:
$$
\Gamma^{\alpha}_{\beta\gamma,\mu} = -\frac{1}{3}(R^\alpha_{\beta\gamma\mu} + R^\alpha_{\gamma\beta\mu})
$$
But the only thing I can get (following the textbook's hint), is:
$$
\Gamma^{\alpha}_{\beta\gamma,\mu} = -\frac{1}{2}R^\alpha_{\gamma\beta\mu}
$$
For the next step I tried to use Riemann tensor symmetries, such as:
$$
R^\alpha_{\beta\gamma\delta} = R^\alpha_{\beta[\gamma\delta]} 
$$
and
$$
R^\alpha_{[\beta\gamma\delta]} = 0
$$
But they didn't bring me to the answer.
 A: I'll use the same notations as yours.
In Riemann normal coordinates, the point is $x^{\alpha}=\lambda\, n^{\alpha}$, the tangent vector is $u^{\alpha} = n^{\alpha}$, the deviation vectors are $N_{(\mu)}^{\alpha} = \lambda \,\delta_{(\mu)}^{\alpha}$.
The geodesic deviation equation is
\begin{equation}\tag{1}
    \frac{\nabla^2 N_{(\mu)}^{\alpha}}{d\lambda^2} = {R^{\alpha}}_{\beta\gamma\delta} u^{\beta} u^{\gamma}  N_{(\mu)}^{\delta}.
\end{equation}
Expanding Eq. (1) using the definition of covariant derivative
\begin{equation}\tag{2}
    \frac{\nabla N_{(\mu)}^{\alpha}}{d\lambda} = \frac{dN_{(\mu)}^{\alpha}}{d\lambda} + \Gamma^{\alpha}_{\beta\gamma} u^{\beta} N_{(\mu)}^{\gamma},
\end{equation}
and the geodesic equation $d u^{\alpha}/d\lambda + \Gamma^{\alpha}_{\beta\gamma} u^{\beta} u^{\gamma} = 0$, we obtain
\begin{equation}\tag{3}
    \frac{d^2 N_{(\mu)}^{\alpha}}{d\lambda^2} + 2 \Gamma^{\alpha}_{\gamma\delta} u^\gamma \frac{dN_{(\mu)}^{\delta}}{d\lambda} + \Bigl( \Gamma^{\alpha}_{\gamma\delta,\beta} - \Gamma^{\alpha}_{\mu\delta} \Gamma^{\mu}_{\beta\gamma} + \Gamma^{\alpha}_{\beta\mu} \Gamma^{\mu}_{\gamma\delta} - {R^\alpha}_{\beta\gamma\delta}  \Bigr) u^{\beta} u^{\gamma}  N_{(\mu)}^{\delta} = 0.
\end{equation}
Substituting $n^{\alpha}$ and $N_{(\mu)}^{\alpha}$ into the above equation, we find
\begin{equation}\tag{4}
    2 \Gamma^{\alpha}_{\gamma\delta} n^{\gamma} + \lambda \Bigl( \Gamma^{\alpha}_{\gamma\delta,\beta} - \Gamma^{\alpha}_{\mu \delta} \Gamma^{\mu}_{\beta\gamma} + \Gamma^{\alpha}_{\beta\mu} \Gamma^{\mu}_{\gamma\delta} - {R^{\alpha}}_{\beta\gamma\delta} \Bigr) n^{\beta} n^{\gamma} = 0.
\end{equation}
We expand $\Gamma^{\alpha}_{\gamma\delta}$ in powers of $\lambda$:
\begin{equation*}
    \Gamma^{\alpha}_{\gamma\delta} = \Gamma^{\alpha}_{\gamma\delta} \big|_{\mathcal{P}_0} + \lambda \Gamma^{\alpha}_{\gamma\delta,\mu} u^{\mu} + O(\lambda^2) = \lambda \Gamma^{\alpha}_{\gamma\delta,\beta} n^{\beta} + O(\lambda^2).
\end{equation*}
Substituting this expression back into (4), dividing through by $\lambda$ and then evaluating on $\mathcal{P}_0$, we arrive at
\begin{equation*}
    \bigl( 3 \Gamma^{\alpha}_{\gamma\delta,\beta} - {R^{\alpha}}_{\beta\gamma\delta} \bigr)\big|_{\mathcal{P}_0} n^{\beta} n^{\gamma} = 0.
\end{equation*}
After symmetrizing in the indices $\beta$ and $\gamma$, we get
\begin{equation}\tag{5}
    \bigl( \Gamma^{\alpha}_{\gamma\delta,\beta} + \Gamma^{\alpha}_{\beta\delta,\gamma} \bigr) \big|_{\mathcal{P}_0} = \frac{1}{3} \bigl( {R^{\alpha}}_{\beta\gamma\delta} + {R^{\alpha}}_{\gamma\beta\delta} \bigr) \big|_{\mathcal{P}_0}.
\end{equation}
By cycling the indices $(\gamma\delta\beta)$ to get two other equations $A$ and $B$, adding $A$ to and substracting $B$ from (5), we finally get
\begin{equation}\tag{6}
    \Gamma^{\alpha}_{\beta\delta,\gamma} \big|_{\mathcal{P}_0} = \frac{1}{3} \bigl( {R^{\alpha}}_{\beta\gamma\delta} + {R^{\alpha}}_{\delta\gamma\beta} \bigr) \big|_{\mathcal{P}_0}.
\end{equation}
If you like, you may also write it as
\begin{equation}\tag{7}
    \Gamma^{\alpha}_{\beta\gamma,\delta} \big|_{\mathcal{P}_0} = - \frac{1}{3} \bigl( {R^{\alpha}}_{\beta\gamma\delta} + {R^{\alpha}}_{\gamma\beta\delta} \bigr) \big|_{\mathcal{P}_0}.
\end{equation}
