What causes geodesic incompleteness in spacetime manifolds?

[dapias09]

Hi all,

I would like to know if somebody know the cases when we have in the space time manifold (and in general in any manifold) geodesic incompleteness.

I know that a case can be a singularity in the curvature scalar (or in general, a singularity in any component of the Riemann tensor). Also when we remove a point of our manifold.

I ignore if when we have a singularity due to a bad selection of coordinates (coordinate singularities), there is geodesic incompleteness. Also ignore more cases.

Thank you in advance.

 

[bcrowell]

Also when we remove a point of our manifold.

I think it's not enough to remove a point from an otherwise well-behaved spacetime, because we're typically only interested in cases where the spacetime can't be extended in order to restore geodesic completeness. But for example, you could have a conical singularity, in which case geodesics can't be extended through the point where the singularity occurs, and this can't be fixed by extending the spacetime.