The context: I'm trying to show that any geodesic normal coordinates in Minkowski spacetime form an inertial coordinate system. I believe it should be the case that on any pseudo-Riemannian manifold which is flat (i.e. the Riemann tensor vanishes), geodesic normal coordinates (exponentiated from an orthonormal basis) should be inertial, i.e. have the metric tensor be diagonal/constant. But I cannot think of a simple proof of this seemingly intuitive statement.
Any ideas would be appreciated, as would any counterexamples, if my claim is false.