Definition:Locally Minimizing Admissible Curve

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.