Does there exist a complete, non-compact Riemannian manifold $(M^n, g)$, which has only one ray starting from one fixed point $p\in M^n$?



Rays on non-compact complete Riemannian manifolds
Recall that a  *ray* is a geodesic defined on $[0,\infty)$ that minimizes the distance between any of its points.


A capped round cylinder has required properties: From any point sufficiently far away from the cap there is only one ray going along the axis of the cylinder. (This is because no ray passed through the cap, and rays on a cylinder are easy to analyse). 
It is also easy to get the same conclusion at each point not on the cap. 


In general, deciding when a geodesic is a ray is nontrivial even for surfaces of revolution. These matters are discussed at length in <http://arxiv.org/abs/1108.1515>. 


