Let $(M,m)$ be a connected Riemannian manifold. In this lecture we turn $M$ into a metric space by declaring that

$$d_m(x,y) := \inf \left\{ \mathbb{L}_m( \gamma) \mid \gamma \in \mathcal{P}_{xy} \right\},$$

where $\mathcal{P}_{xy}$ denotes the set of all piecewise smooth maps $\gamma \colon [a,b] \to M$ such that $\gamma(a) = x$ and $\gamma(b) = y$. We prove moreover that the metric space topology $d_m$ induces on $M$ coincides with its original topology. We also show that any geodesic is locally length-minimising with respect to $d_m$ (which explains the name “geodesic”).

Next lecture we will prove that $(M , d_m)$ is complete as a metric space if and only if $m$ is complete as a metric (i.e. all geodesics are defined on the entire real line). This is the Hopf-Rinow Theorem.