Definition:Riemannian Arc Length Function of Admissible Curve

Definition

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.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions