Surface where Hopf Rinow's theorem is not worth

Question

In Hopf Rinow's Theorem we see that if $S$ is a complete surface then we have: for all $p,q \in S$ there is $\gamma$ a minimizing geodesic that connects them. However, as long as the reciprocal of this result does not occur, this may be true even if $S$ is not complete. So I would like to ask if anyone knows a surface example (obviously not complete) where there are two $p,q$ points, such that there is no minimizing geodesic that connects them. I thought of the cone without the vertex, but couldn't find such points. Thank you for your help

Answer 1

Let $S$ be the punctured plane $\mathbb{R}^2 / \{0\}$ and let $x,y$ be two points on one of the axis but on different semiplanes, let us say $(1,0)$ and $(-1,0)$. Obviously no minimizing geodesic can join $x$ and $y$.