Normal coordinates

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In Riemannian geometry, the normal coordinates at p consist of a chart such that locally the symmetric part of the Christoffel symbols vanish, ie $\Gamma^a_{(bc)} =0$ . Furthermore, at p, the following equations hold

$g_{ij}(p) = \delta_{ij}, \quad \frac{\partial g_{ij}}{\partial x^k}(p) = 0, \quad \Gamma^i_{jk}(p) = 0.$

Therefore, the covariant derivative reduces to a partial derivative, and the geodesics through p are locally linear functions of t. This idea was implemented by Einstein in his General Relativity using his Equivalence Principle and understanding the normal coordinates as an inertial frame.

Normal coordinates are specific to Riemannian geometry. They do not generalize to Finsler geometry (Rund, 1959).