Riemannian distance on a connected manifold

On a connected Riemannian manifold define $d_g(p,q)=\inf \{L_g(\gamma ):\gamma \text{ is piecewise }{C}^1\text{ from }p\text{ to }q\}$.

Lengths are those of Riemannian speed and length. For each pair $p,q$, Any two points in a connected smooth manifold can be joined by a piecewise c one curve supplies a curve, so the set of lengths is nonempty, contains a finite real number and is bounded below by zero. Applying the least-upper-bound property The Cauchy-sequence reals have the least-upper-bound property to the negatives gives a finite nonnegative infimum. On the empty connected manifold this defines the empty distance function; there are no pairs to evaluate. No minimizing curve is part of this definition.

Lee, Chapter 13, pp.337–340, Proposition 13.25, Lemma 13.28 and Theorem 13.29; finite piecewise $C^1$ refinements and pauses are treated explicitly here.