Theorem
Given a connected Riemannian manifold, it is a geodesically complete manifold if and only if it is a Cauchy complete metric space.
$\blacksquare$
That is, if we have that if the geodesics can be extended to all of R (with a constant speed parametrization) then every pair of points can be joined by a curve whose length is the minimum of the lengths of the curves joining them.
Counterexample: let $S$ be the punctured $xy$-plane,
$$S := \{\,(x, y, 0) : (x, y) \neq (0, 0)\,\}. $$
Then, there is no smooth geodesic in $S$ connecting, say, $(-1, 0, 0)$ to $(1, 0, 0)$.
Remark
There is no Hopf-Rinow theorem for pseudo-Riemannian manifolds.
$\blacksquare$
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Author of the notes: Antonio J. Pan-Collantes
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