# Christoffel Symbols vanish at Origin of Normal Neighborhood

## Theorem

Let $\struct {M, g}$ be an $n$-dimensional Riemannian or pseudo-Riemannian manifold.

Let $U_p$ be the normal neighborhood for $p \in M$.

Let $\struct {U_p, \tuple {x^i}}$ be a normal coordinate chart.

Let $\set {\Gamma^i_{jk}}$ be Christoffel symbols.

Then:

$\map {\Gamma^i_{jk} } {\map {x^r}p} = 0$