The geodesics are straight lines, in geodesic normal coordinates, just when the associated projective connection is flat. See Kobayashi and Nagano, On projective connections, Journal of Mathematics and Mechanics, vol. 13, no. 2, 1964. I am not sure that I have seen a precise description of this condition in terms of the curvature, although there must be one. (I mistakenly thought it was the vanishing of the Weyl tensor, but that describes conformal, not projective, flatness.)
Ben McKay
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