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By definition, a geodesic is a mapping $\gamma: I = (0, 1) \rightarrow M$ such that $\nabla_{\dot{\gamma}(t)} \dot{\gamma}(t) = 0$.

Here we only need the connection. So, we do not need a metric to define a geodesic? Is such kind of geodesic useful?

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    $\begingroup$ I believe such a geodesic is commonly called affine geodesic. $\endgroup$ – Loreno Heer Aug 25 '15 at 12:58
  • $\begingroup$ Seems like having a non-metric connection, one can define a parallel transport and then geodesics. I am not en expert, you may check what is done with Cartan connection. $\endgroup$ – A.Γ. Aug 25 '15 at 13:25
  • $\begingroup$ Yes, geodesics are used in affine geometry; a good example to think about is of a pseudo-Riemannian manifold. It does have connections, geodesics, but no compatible Riemannian metrics. $\endgroup$ – Moishe Kohan Aug 25 '15 at 20:55

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