Definition:Unit-Speed Curve

Definition

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

Let $I$ be a closed real interval.

Let $\gamma : \R \to M$ be a smooth curve.

Suppose the speed of $\gamma$ is equal to $1$:

$\size {\map {\gamma'} t}_g = 1$

where $\size {\, \cdot \,}_g$ is the Riemannian inner product norm.

Then $\gamma$ is a unit-speed curve.

Sources

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