Suppose I have a global coordinate system on a manifold, which is affine with respect to an affine connection on that manifold. The connection is flat and torsion free, and the connection coefficients are 0 w.r.t to the coordinate system. What additional properties of the connection would result in all geodesics being straight lines in the coordinate system? What about only locally? Note that the connection is not necessarily a Levi-Civita connection. I'm new to differential geometry so please excuse any mistakes in terminology.
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3$\begingroup$ If the connection coefficients are zero, just write down the geodesic equation: $\ddot{x}+\Gamma \dot{x} \dot{x}=0$, with $\Gamma=0$, to get $\ddot{x}=0$, i.e. no acceleration, i.e. constant velocity, i.e. linear motion. $\endgroup$– Ben McKayCommented Mar 8, 2018 at 7:12
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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.
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1$\begingroup$ NB the linked Wikipedia articles are about the conformal Weyl and Cotton tensors, not the projective Weyl and Cotton tensors. One can find definitions of the latter, e.g., in arxiv.org/pdf/1003.1469.pdf . $\endgroup$ Commented Mar 10, 2018 at 11:34