42. 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 $ \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 $T M$.
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 $d_m$.
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