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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.)

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. If an affine connection is projectively flat, then the Weyl and Cotton tensors vanish, as these are projective connection invariants. In dimensions 3 or higher, these conditions force the affine connection to be that of a constant curvature Riemannian metric. Indeed, Beltrami proved that a projectively flat affine connection is locally that of a constant curvature Riemannian metric.

The geodesics are straight lines, in geodesic normal coordinates, just when the associated projective connection is flat, i.e. the Weyl tensor vanishes. See Kobayashi and Nagano, On projective connections, Journal of Mathematics and Mechanics, vol. 13, no. 2, 1964.

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.)

The geodesics are straight lines, in geodesic normal coordinates, just when the associated projective connection is flat, i.e. the Weyl tensor vanishes. See Kobayashi and Nagano, On projective connections, Journal of Mathematics and Mechanics, vol. 13, no. 2, 1964.

The geodesics are straight lines, in geodesic normal coordinates, just when the associated projective connection is flat, i.e. the Weyl tensor vanishes. See Kobayashi and Nagano, On projective connections, Journal of Mathematics and Mechanics, vol. 13, no. 2, 1964.