Problem
Problem 6.54
(i) Find an example of a connected Riemannian manifold $(M, g)$ to show that the property "Any $p, q \in M$ can be joined by a geodesic whose arc length equals the distance $d_{g}(p, q)$ " (see Problem 6.53) does not imply that $M$ is complete. (ii) Find an example of a connected Riemannian manifold to show that a minimal geodesic between two points need not be unique; in fact, there may be infinitely many.