Definition:Riemannian Length of Admissible Curve

Definition

Let $\struct {M, g}$ be a Riemannian manifold.

Let $\closedint a b$ be a closed real interval.

Let $\gamma : \closedint a b \to M$ be an admissible curve.

Then the (Riemannian) length of $\gamma$, denoted by $\map {L_g} \gamma$, is defined by:

$\ds \map {L_g} \gamma := \int_a^b \size {\map \gamma t}_g \rd t$

where $\size {\map \gamma t}_g$ is the Riemannian inner product norm.