Naively I would have thought that a manifold becomes geodesically incomplete if there are missing points in it or if the geodesics are hitting a boundary.
But I am not sure how to think of geodesic completeness or not for manifolds with a boundary. Like how to think for the closed disk on $\mathbb{R}^2$
I would like to know of other generic ways in which geodesic incompleteness can happen. (Like for pseudo-Riemannian manifolds the Hawking-Penrose theorems pin down many generic scenarios)
Like from $\mathbb{R}^2$ (with the standard metric) if one removes the point $(0,0)$ then there is no geodesic from say the point $(-1,0)$ to $(1,0)$.
- But I can't visualize how I can actually put a complete metric on this punctured plane and make it geodesically complete? I can try to make the metric hyperbolic near the deleted point so that the geodesics approaching it never actually reach it but go off to infinity but even then I can't see how it will produce a geodesic which connects $(-1,0)$ to $(1,0)$
Also there are two other ways of producing geodesic incompleteness which I know but don't have a good understanding of,
Any open subset of a complete connected Riemannian manifold is geodesically incomplete. (I guess this follows from Hopf-Rinow but I can't see it clearly)
A noncompact surface which is not diffeomorphic to $\mathbb{R}^n$, and if for some metric every point on this surface has positive curvature, then the metric on it must be incomplete.
I would be happy to see explanations about the above things and also examples for the second case.
Also is it possible to write down as formulas simple examples of incomplete geodesics?