Consider infinitely many distinguishable observers, no two of whom ever meet; and who generally "keep sight of each other", but not necessarily "each keeping sight of all others".
How should they determine whether or not they can be described as being "defined on a Lorentzian manifold"?
[This question refers to terminology of http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity and is meant as follow-up to that question; in the attempt to ask perhaps more originally.]
The phrase "defined on a Lorentzian manifold" appears a very general condition.
To be more specific consider instead the question:
"How should the given observers determine whether or not some subset of the entirety of events in which they (separately) participated can be described as "open set of a 3+1 dimensional Lorentzian manifold"?