The asker is pretty clearly asking:
"Presumably the easiest method of checking that M is globally hyperbolic is to find a Cauchy surface and somehow prove that it is a Cauchy surface. How does one actually do this?" [emphasis mine] This indicates the asker is well-aware that to prove X, you should check that the definition of X applies, but is unable to do so in specific cases. —
ACuriousMind 15 mins ago