# Fermi coordinates

This article defines a property that is evaluated for a choice of local coordinates at a point on a Riemannian manifold

## Definition

Let $(M,g)$ be a Riemannian manifold. Let $p \in M$ and $\gamma$ a geodesic curve through $p$. Then a choice of local coordinates at $p$ is said to give Fermi coordinates if:

• For small changes of the first variable, we get the geodesic in the neighbourhood of $p$. In other words, the geodesic locally is the axis for the first coordinate
• The metric tensor, restricted to that geodesic, is the Euclidean metric
• All Christoffel symbols vanish on the geodesic