# 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 be a Riemannian manifold. Let and a geodesic curve through . Then a choice of local coordinates at is said to give Fermi coordinates if:

- For small changes of the first variable, we get the geodesic in the neighbourhood of . 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