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  "body": "\n## Summary\n\nNormal coordinates at a point $p$ of a [[Riemannian manifold|7CBT4SM83Y]] are local coordinates centred at $p$.\nThey identify a neighbourhood of $p$ with a ball in the tangent space through the [[exponential map|Y7Z0AGY1A7]].\nIn them, the metric equals the Euclidean metric at $p$ and its first derivatives vanish at $p$, so the [[Christoffel symbols|FZBA1HA9ZM]] vanish at $p$.\nThey make many local computations, such as those of curvature identities, much simpler.\n\n## Explanations\n\n### L1 human\n\nNear a point of a curved space, normal coordinates are the best possible imitation of flat coordinates.\nEach nearby point gets the coordinates of the direction and distance one walks along a geodesic to reach it from $p$.\nStraight lines through the origin in these coordinates are exactly the geodesics through $p$.\nOn the Earth, this is like the azimuthal equidistant map centred at one city, which gets distances and directions from that city right.\n",
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