Normal coordinates

Summary

Normal coordinates at a point $p$ of a Riemannian manifold are local coordinates centred at $p$. They identify a neighbourhood of $p$ with a ball in the tangent space through the exponential map. In them, the metric equals the Euclidean metric at $p$ and its first derivatives vanish at $p$, so the Christoffel symbols vanish at $p$. They make many local computations, such as those of curvature identities, much simpler.