Unit-Speed Admissible Curve is Critical Point of Riemannian Length iff Geodesic

Theorem

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

Let $\gamma$ be a unit-speed admissible curve.

Let $L_g$ the Riemannian length of some admissible curve.

Then $\gamma$ is the critical point of $L_g$ if and only if $\gamma$ is 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