If m is any point in the Riemannian manifold M, then there exists a local coordinate system at m such that:

We call such a coordinate system a **local inertial frame** or a **normal frame.**

**Corollary: **As all the first order derivatives of the metric are null, given the Christoffel symbol expression:

then in a local inertial referential the vanishing of the partial derivatives of the metric tensor at any point of M **is equivalent to the vanishing of Christoffel symbols** at that point and in this referential **the geodesics are straight lines**.

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