43. Geodesics and Sprays
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 $ \dot \gamma$ is parallel with respect to $ \nabla$. We prove that for any $(p,\xi) \in T M$, there exists a geodesic $ \gamma$ with $ \gamma(0) = p$ and $\dot \gamma(0) = \xi$, 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 $T M$.
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 $g$ on $M$, then later in the course we will see that (a) $g$ determines a metric $d_g$ (in the sense of topology) and (b) geodesics with respect to $\nabla$ are indeed locally length-minimising with respect to $d_g$.
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