Equivalent ways to refer to Lorentzian geometry

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I'm a curious student about the Yamabe-type problem in Lorentzian geometry. I found a single article about this topic (2015).

I don't know much about Lorentzian geometry and I wonder if one may refers to this subject in a different way. For example, are Minkowsky spaces or hyperbolic manifolds equivalent to Lorentzian manifolds? I mean, articles that solve hyperbolic Yamabe problem or treat Minkowsky conformal invariants are related to the Yamabe problem in Lorentzian manifolds?

Could you give me a brief explanation about those spaces and the difference between them? Also, if someone is aware of recent progress in this area, it would be interesting.

Thank you in advance.