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I know the Hopf-Rinow theorem and that if a Riemannian manifold is complete it implies that given any two points there is a unique distance minimizing geodesic that connects the two points, but is the converse true? Given that a Riemannian manifold is such that any two points has a length minimizing geodesic, is it complete?

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No; consider as a counterexample the manifold $(0,1) \subseteq \mathbb{R}$, equipped with the metric it inherits from the Euclidean metric on $\mathbb{R}$.

I might call such a Riemannian manifold (geodesically) convex (though this sometimes carries the stronger implication that any two points are connected by a unique minimizing geodesic). I think this terminology gives the right picture.

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