Converse of statement related to Hopf-Rinow Theorem

Question

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?

Answer 1

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.