Definition:Locally Minimizing Admissible Curve
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Definition
Let $\struct {M, g}$ be a Riemannian manifold.
Let $I = \closedint c d$ be a closed real interval.
Let $\gamma : I \to M$ be an admissible curve.
Suppose for all $t_0 \in I$ there exists a neighborhood $I_0 \subseteq I$ such that if $a, b \in I_0$ and $a < b$ then the restriction $\gamma \restriction_{\closedint a b}$ is minimizing.
Then $\gamma$ is said to be locally minimizing.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Geodesics Are Locally Minimizing