Riemann normal coordinates
This article defines a property that is evaluated for a choice of local coordinates at a point on a Riemannian manifold
A system of local coordinates at a point in a Riemannian manifold is said to be Riemann normal coordinates if the Christoffel symbols with respect to those coordinates, are all zero at that point. In other words, the Levi-Civita connection is zero in that coordinate system.
The notion of Riemann normal coordinates makes sense in the greater generality of a differential manifold equipped with a connection. In this case, we simply require that the coefficients of that connection should vanish at the point.