The most recent textbook answer I could find would be from Choquet-Bruhat's 2009 book "General Relativity and the Einstein Equations". She writes on page 403:

**Remark** A curvature singularity does not imply geodesic incompleteness. The geodesic flow depends only on the $C^{1,1}$ structure of the metric. Conversely, does geodesic incompleteness imply a curvature singularity? This question is linked with the strong cosmic censorship conjecture defined in the previous chapter. ...

As the strong cosmic censorship conjecture is still - as far as I know - a conjecture, I'd say there is no such extension of these theorems, so far. However, I remember vaguely that there are a lot of different formulations for the cosmic censorship conjectures. Some of the stronger ones are, iirc, disproven. The weaker you get, the more open the question is.

Along similar lines was what I found in the slightly older book by Kriele, "Spacetime", roughly its chapter 9.

Regarding yess' comment below:
I can only give more ressources, when it comes to pathological behaviour of the curvature at accessible events. In Hawking, Ellis on page 290ff there is an example involving Taub-NUT which I don't follow completely. Then, Curiel wrote both an article on plato and a paper along the same lines, which can be accessed on his website. The bibliography seems to be a good starting point for further literature search. For example, the "marketing excerpt" of Clarke's "The Analysis of Space-Time Singularities" sound already very promising - sadly I cannot get my hands on it. This paper by Ellis and Schmidt has some interesting examples in section 4 on non-scalar singularities. If I understand the Clarke excerpt, page 7, correctly, it can be traced back to matters of regularity - like the above remark of Choquet-Bruhat also alludes to.

This post imported from StackExchange Physics at 2015-07-26 09:36 (UTC), posted by SE-user Wraith of Seth