geodesic based on fixed points

Question

Question: For any points $p,q\in M$, does there exist a geodesic curve connecting $p$ and $q$?

• Let $M$ be some constant curvature space, like $\mathbb R^n$, $\mathbb S^n$, $\mathbb H^n$. The answer is yes in $M$.

• However, an obvious counterexample is considering that $p,q$ are in different connected components.

My point is that the question is not generally right. But I need a counterexample which $M$ is connected. Any advice is helpful. In addition, the geodesic curve can be piecewise differentiable.

Answer 1

As Ted's example shows, you need some extra hypothesis like completeness. Part of the theorem of Hopf and Rinow is that under this hypothesis every pair of points is joined by a geodesic (and the geodesic can be chosen to realise the distance between the two points)

In the compact case, a famous theorem of Serre tells us that not only is there a geodesic joining two points but many. See https://mathoverflow.net/questions/135355/proof-of-a-theorem-of-jean-pierre-serre-on-geodesics-of-closed-riemannian-manifo

Answer 2

Try $\Bbb R^2-\{0\}$ with the standard metric (or any incomplete Riemannian manifold). There is no geodesic path of length equal to the distance between two antipodal points. If you're going to allow broken geodesic paths, I doubt there's a counterexample.