Q:

Global Hyperbolicity in spacetime Manifold

If space time is timelike or null geodesically incomplete but cannot be embedded in a larger spacetime then we say that it has singularity. What does incompleteness means here?

A:

A manifold $M$ is geodesically complete (i.e. has the property of "geodesic completeness") if for every point $P\in M$ and every direction $dx^\mu$ from that point, the infinitesimal line interval $dx^\mu$ away from $P$ may be transported by parallel transport by any amount $K\in(-\infty,+\infty)$ in both directions, i.e. if the geodesics (maximally straight lines in the manifold) may be continued indefinitely.
If that's not the case, there exists a maximal $K_0$ above (or minimum $-K_1$ below) which the prolongation of the geodesic (i.e. further continuation of the parallel transport) is impossible. This "stop sign" may either be averted by embedding the manifold in a larger one (the old one basically "was" just a cut taken from a larger, geodesically complete, manifold), or no extension is possible.
The only reason why the extension may be impossible is that something "bad" is happening at the point of the geodesic (the end point) corresponding to $K_0$, i.e. the curvature invariants go to infinity. That's "something bad" is known as the "singularity". The very point $K_0$ must already be removed from the original manifold because the neighborhood of this point isn't diffeomorphic to a patch of $R^N$. And because even the singular point $K_0$ must be outside the manifold, it's clear that we can't continue past $K_0$, either.

