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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.
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Geodesics