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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. 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.