= Riemannian distance
{c}
{title2=$d_g$}
The distance on a <connected> <Riemannian manifold> is the <infimum> of the <arc lengths> of piecewise smooth <curves> joining the endpoints:
$$
d_g(p,q)=\inf_{\gamma:p\leadsto q}L_g(\gamma).
$$
It answers “how far must I travel while staying in this geometry?” The <Riemannian metric> measures an instantaneous <tangent vector>, whereas <Riemannian distance> is the global result of comparing complete routes.
In <Euclidean space> the straight segment achieves this <infimum>, giving $d(p,q)=\|q-p\|$. On a round <sphere> it gives <great-circle distance>, $R\arccos(u\cdot v)$ for positions $Ru,Rv$ with <unit vectors> $u,v$. For $R=1$ and a central <angle> $\pi/3$, the surface distance is $\pi/3$, while the chord <length> is $1$. On a <circular cylinder>, routes with different winding lead to the minimum in <Riemannian distance on a circular cylinder>.
The word <infimum> matters: a shortest route need not exist. In the Euclidean plane with the origin removed, the points $(-1,0)$ and $(1,0)$ still have <Riemannian distance> $2$, approached by paths detouring around an arbitrarily small circle. Every allowed route has <length> strictly greater than $2$, since the only Euclidean straight segment passes through the missing point. The <Hopf-Rinow theorem> guarantees a <minimizing geodesic> between any two points when a <connected> <Riemannian manifold> is complete.
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