I have two 2-dimensional space-times. One of them is flat the other one has not-vanishing curvature (Riemann tensor). But they seem to have a similar global and causal structure. Of course, because of the 2-dimensional case they are local conformally flat. I am looking for a relation between them that could explain the similar causal structure and I think that a conformal transformation would be nice. 1) How do I know (prove) whether there exists a (global) conformal transformation between them? Is there a way to prove that there exists one (I do not need the transformation mapping itself, only the proof of existence)? 2) Are there other global properties of space-times that are worth discussing? What are the typical global properties (my guess: Horizons, causal light cone structure...what else)?