---
tag: RRHKMGNKG9
kind: concept
name: Normal coordinates
msc: [53B20, 53C22]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_coordinates
same_as:
  wikidata: Q2152232
  wikipedia: Normal coordinates
---

## Summary

Normal coordinates at a point $p$ of a [[Riemannian manifold|7CBT4SM83Y]] are local coordinates centred at $p$.
They identify a neighbourhood of $p$ with a ball in the tangent space through the [[exponential map|Y7Z0AGY1A7]].
In them, the metric equals the Euclidean metric at $p$ and its first derivatives vanish at $p$, so the [[Christoffel symbols|FZBA1HA9ZM]] vanish at $p$.
They make many local computations, such as those of curvature identities, much simpler.

## Explanations

### L1 human

Near a point of a curved space, normal coordinates are the best possible imitation of flat coordinates.
Each nearby point gets the coordinates of the direction and distance one walks along a geodesic to reach it from $p$.
Straight lines through the origin in these coordinates are exactly the geodesics through $p$.
On the Earth, this is like the azimuthal equidistant map centred at one city, which gets distances and directions from that city right.