# When are geodesics straight lines?

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.

• 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. – Ben McKay Mar 8 '18 at 7:12