Hopf-Rinow theorem

Theorem
Given a connected Riemannian manifold, it is a geodesically complete manifold if and only if it is a Cauchy complete metric space.
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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,

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.
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