\begin{definition}[Definition:Riemannian Arc Length Function of Admissible Curve]
Let $\struct {M, g}$ be a Riemannian manifold.
Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $\gamma : \closedint a b \to M$ be an admissible curve.
The '''(Riemannian) arc length function of $\gamma$''' is the real function $s : \closedint a b \to \R$ defined by:
:$\ds \map s t := \map {L_g} {\valueat \gamma {\closedint a t} } = \int_a^t \size {\gamma}_g \rd t$
where $L_g$ is the Riemannian length of $\gamma$, and $\size {\, \cdot \,}_g$ is the Riemannian inner product norm.
\end{definition}