I need to find an example of a connected riemannian manifold $(M,g)$ and a point $p \in M$ such that the exponential map $\exp_p : T_pM \to M$ is well-defined, but is not surjective.
Taking $\mathbb{R}^n$ and removing a point won't work because the resulting space won't be geodesicly complete and $\exp_p$ won't be defined over whole tangent space $T_p M$.
Every compact riemannian manifold will have surjective $\exp_p$. So, I can focus only on non-compact manifolds
Here the first idea is to construct a surface similar to $\mathbb{R}^2$ but with a hole such that a geodesic which will normally connect two points from the different sides of the hole, will run down the hole (possibly to another side of the surface). The problem is that there are many such surfaces, for example, one can b e defined parametricly as
$$
r(x,y) = \Big(x,y, \ln\sqrt{x^2 + y^2}\Big)
$$
(this one doesn't have "running to the other side" effect, by the way). But it still very hard for me to write explicitly equations for geodesics and analyse there behavior near the hole.
So, I need an example of a manifold or a surface where the rigorous proof for $\exp_p$ not being surjective can be carried out with a relative ease.
Best Answer
There is no such example. The typical proof of the Hopf-Rinow theorem shows that if $\exp_p$ is defined on all of $T_pM$ for some $p\in M$, then $M$ is metrically complete, and in that case, every $q\in M$ can be joined to $p$ by a distance-minimizing geodesic $\gamma$. If we set $v = \gamma'(0)\in T_pM$, we then have $q = \exp_p(tv)$ for some $t\in \mathbb R$.
See Corollaries 6.20 and 6.21 in my Introduction to Riemannian Manifolds (2nd ed.).