Let $\nabla$ be a connection on (the tangent bundle of) a smooth manifold $M$. A geodesic $\gamma$ is a smooth curve in $M$ such that $\gamma’$ is parallel with respect to $\nabla$. We prove that for any $(x,v) \in T M$, there exists a geodesic $\gamma$ with $\gamma(0) = x$ and $\gamma’(0) = v$, which moreover is unique up its domain of definition.

We then explain how geodesics can be seen projections to $M$ of integral curves of sprays, which are a special class of vector fields on $T M$.

##### Warning

Geodesics are usually thought of as being “shortest paths”. In the general setting we study in this lecture this need not be the case! However if $\nabla$ is the Levi-Civita connection associated to some Riemannian metric $m$ on $M$, then in Lecture 52 that (a) $m$ determines a metric $d_m$ (in the sense of topology) and (b) geodesics with respect to $\nabla$ are indeed locally length-minimising with respect to $d_m$.