# Geodesics without a metric

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?

• I believe such a geodesic is commonly called affine geodesic. – Loreno Heer Aug 25 '15 at 12:58
• 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. – A.Γ. Aug 25 '15 at 13:25
• 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. – Moishe Kohan Aug 25 '15 at 20:55