Admissible Curve in Riemannian Manifold has Unique Forward Reparametrization by Arc Length

Theorem

Let $\struct {M, g}$ be a Riemannian manifold with or without boundary.

Let $I \subseteq \R$ be a real interval.

Let $\gamma : I \to M$ be an admissible Curve.

Then $\gamma$ has a unique forward reparametrization by arc length.

Proof

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Sources

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