Geodesic between two points I have a question about geodesics.
So far I know that for any surface $S$ defined by some immersion $f: U \subset\mathbb{R}^2 \rightarrow S \subset \mathbb{R}^3,$ we have that for any point on the surface and any direction, there exists locally a geodesic ( due to Picard-Lindelöf applied to the geodesic equation). 
But what can we say globally? To me it would be more natural to ask that if our surface is path-connected, does this imply that there exists a geodesic between any two points? What about more general manifolds, is this then true? Is there a similar existence theorem for the geodesic ODE in the context of this boundary value problem?

Given a surface $S \subset \mathbb{R}^3$ defined by a smooth immersion as indicated, there need not be a smooth geodesic connecting a given pair of points.
For example, let $S$ be the punctured $xy$-plane,
$$S := \{\,(x, y, 0) : (x, y) \neq (0, 0)\,\}. $$
Then, there is no smooth geodesic in $S$ connecting, say, $(-1, 0, 0)$ to $(1, 0, 0)$.
