Riemannian speed and length

The Riemannian speed on a $C^1$ piece is $\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}_g=\sqrt{g_{\gamma (t)}(\dot{\gamma }(t),\dot{\gamma }(t))}$. Its length is $L_g(\gamma )={\sum }_j{\int }_{t_{j-1}}^{t_j}\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}_{g}dt$.

The curve convention is Piecewise c one curve on a manifold and the norm is Pointwise norm and angle from a riemannian metric. Each integrand is continuous on its closed piece with the one-sided endpoint derivative, hence Riemann integrable and nonnegative. Values chosen at the finitely many corners do not change its integral. For a singleton interval the empty sum is zero; a constant curve also has zero length. Partition independence is established by Riemannian length is independent of piecewise c one subdivision ↗.

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.