Unit-Speed Minimizing Curve is Geodesic

Theorem

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 curve.

Suppose $\gamma$ is parametrized so that $\gamma$ is a unit-speed curve.

Suppose $\gamma$ is minimizing.

Then $\gamma$ is a geodesic.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Minimizing Curves Are Geodesics