Existence of a local geodesic frame Let $(M,g)$ be a Riemannian manifold of dimension $n$ with Riemannian connection $\nabla,$ and let $p \in M.$ Show that there exists a neighborhood $U \subset M$ of $p$ and $n$ (smooth) vector fields $E_1,...,E_n \in \chi (U),$ orthonormal at each point of $U,$ s.t. at $p,$ $\nabla_{E_i}E_j(p)=0.$
What I've found so far;
Let $V$ be a normal neighborhood at $p.$ Let $(W,x(x_1,...,x_n))$ be a local coordinate system at $p$ s.t. $x(W) \subset V.$  Let $\{X_i:=\frac{\partial}{\partial x_i}\}$ be the standard basis of $T_pM.$ By Gram-Schmidt, there exists an orthonormal basis $\{E_i\}$ of $T_pM.$ Consider the orthonormal frame $E_i(t),$ by parallel transporting $\{E_i(0)=E_i\}$ along a  a geodesic $\gamma :I \to M$ starting at $\gamma(0)=p$ and ending in $\gamma(1)=q,$ where $q \in V.$ This gives us an orthonormal frame at each point of $V.$ To satisfy the second condition, let $E_i(t)=\sum_{l}a_{li}X_l$ and  $E_j(t)=\sum_{s}b_{sj}(t)X_s.$ Then 
$$\nabla_{E_i}E_j=\sum_{k}(\sum_{l,s} a_{li}(t)b_{sj}(t)\Gamma^k_{ls}+E_i(b_{kj}))X_k,$$
where $\Gamma^k_{is}$ are Christoffel symbols. 
To have $\nabla_{E_i}E_j(p)=0,$ we must have $\sum_{l,s}a_{li}(0)b_{sj}(0)\Gamma^k_{ls}+E_i(b_{kj})(p)=0$ for each $k.$ 

How can I conclude the argument?


Recall parallel transport is a isometry, i. e. if $P=P_{\gamma, t_0, t}: T_{c(t_0)}M\rightarrow T_{c(t)}M$ is parallel transport then
$<u, v>_{c(t_0)} = <P(u), P(v)>_{P(c(t_0))}$
then, because you get ${E_i}$ is orthonormal, $<E_i, E_j> = \delta_{i j}$ is the kronecker delta. Therefore $g_{ij} = <E_i, E_j> = \delta_{i j}$ thus $\Gamma_{ij}^k(p) = 0$.
Thereupon $\nabla_{E_i}E_j(p) = \sum_k \Gamma_{ij}^k(p)E_k(p) = 0$.
