Piecewise Smooth Vector Field along Admissible Curve is Variation Field

Theorem

Let $M$ be a smooth manifold.

Let $\gamma$ be an admissible curve on $M$.

Let $V$ be a piecewise smooth vector field along $\gamma$.

Then $V$ is the variation field of some variation of $\gamma$.

Proof

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Sources

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