Definition:Geodesically Complete Semi-Riemannian Manifold

Definition

Let $\struct {M, g}$ be a semi-Riemannian manifold.

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Let $I$ be an real interval.

Suppose every maximal geodesic $\gamma : I \ni t \to M$ is defined for all $t \in \R$.

That is, suppose that every maximal geodesic in $M$ is a geodesic $\gamma : \R \to M$.

Then $M$ is geodesically complete.

Sources

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