As the title says, I am looking for an example of a complete Riemannian manifold $M$ and two points $p,q\in M$ such that:
- $q\in\mathrm{Cut}(p)$, and
- $p,q$ are connected by a unique minimizing geodesic.
I know that in this case, $p,q$ would have to be conjugate points. But I can't think of any simple example.
This answer links to a text that gives the paraboloid as an example (in the context of bifuraction points), but does not really prove it. I would try to do it myself, but the geodesic equations for the paraboloid seem pretty messy. Is there a simpler example, or a clean way to see why the parboloid gives such an example?
