Arc length ($L(\gamma )$)

The arc length of a regular curve $\gamma :[a,b]\to M$ is

For a piecewise smooth curve, $L(\alpha )=\int \Vert {\alpha }^{\prime }(t)\Vert dt$ in an ambient Euclidean space, or the corresponding Riemannian metric norm on a manifold. This is invariant under regular monotone reparametrization.

The distance induced by a connected Riemannian manifold is the infimum of the lengths of piecewise smooth curves joining two points:

In a connected Riemannian manifold, choose a small compact geodesic sphere around $p$ with $0<r<d(p,q)$. Minimize $d(\cdot ,q)$ on this sphere. Every path from $p$ to $q$ crosses it, so an almost-minimizing path has length at least $r+{\min }_{S_r(p)}d(\cdot ,q)$. The triangle inequality gives the reverse bound, yielding the displayed equality. No global completeness is needed.

A minimizing geodesic realizes the Riemannian distance between its endpoints. A geodesic segment is locally minimizing, but need not remain globally minimizing beyond a cut point.

On a complete connected Riemannian manifold, let $c(v)$ for a unit tangent vector $v$ be the supremum of times for which $t\mapsto {\exp }_p(tv)$ minimizes Riemannian distance from $p$. The cut locus consists of the endpoints ${\exp }_p(c(v)v)$ with $c(v)<\infty$. Radial minimization fails beyond this first cut time; distance spheres can cease to be smooth there.