Definition:Maximal Geodesic

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Let $M$ be a smooth manifold.

Let $I, \tilde I \subseteq \R$ be open real intervals.

Let $\gamma : I \to M$ and $\tilde \gamma : \tilde I \to M$ be geodesics.

Suppose there is no $\tilde I$ properly containing $I$ over which $\tilde \gamma$ is restricted to $\gamma$:

$\neg \exists \tilde I : I \subsetneqq \tilde I : \tilde \gamma \restriction_I = \gamma$

Then $\gamma$ is said to be a maximal geodesic.