Definition:Curve Parametrized by Arc Length

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

Let $I := \closedint a b$ be a closed real interval.

Let $\gamma : I \to M$ be an admissible unit-speed curve.

Suppose $a = 0$ and $b > 0$.

Then $\gamma$ is said to be parametrized by arc length.