A while ago I asked this question in Math Stackexchange. Since I didn't receive an answer so far, I thought I'd ask it here.
Suppose $Y$ is a proper length space, where every pair of points $x,y\in Y$ can be joined by a unique length minimizing geodesic (i.e. global geodesic). Can there still be more than one local geodesic joining some two points, or is it necessarily so that the space is also uniquely locally geodesic?
If a uniquely geodesic space which is not uniquely locally geodesic exists, can one even cook up Riemannian manifold as an example?