TITLE: Christoffel symbols vanish in a system of normal coordinates.
QUESTION [7 upvotes]: I'm reviewing for a differential geometry exam and am getting stuck in a proof. This is based on question 4 from section 4-6 from little Do Carmo. 

Show that in a system of normal coordinates centered at $p$, all the Christoffel symbols are zero at $p$.

The question seems fairly routine, but I'm getting stuck in formalizing. Say we have a surface $S$ and $p\in S$. I know there is some parametrization that gives $E=G=1, F=0$ but surely that is not enough to conclude that the Christoffel symbols disappear. I think my difficult in understand is stemming from not fully understanding how our parametrization looks. Any elucidating is thoroughly appreciated.

REPLY [2 votes]: I disagree that knowing $E=G=1$ and $F=0$ is not enough.
I don't recall that Do Carmo provides in explicit formula for the Christoffel symbols, but in a discussion before the Theorema Egregium proof, he implies that such formulae exist. I provide you one here,
$$\begin{bmatrix}E&F\\F&G\end{bmatrix}\begin{bmatrix}\Gamma^u_{uu}\\\Gamma^v_{uu}\end{bmatrix}=\begin{bmatrix}E_u/2\\F_u-E_v/2\end{bmatrix}. $$
By this and the other three formulae, you can conclude every Christoffel symbol vanishes without appeal to geodesics.