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I have split off a stub globally hyperbolic Lorentzian manifold from Cauchy surface
Wow, I didn’t realise that the definition of globally hyperbolic was so simple!
Wow
Yes, it is surprising that this is not advertized more widely.
I have now added pointers to the theorems that establish this.
I have expanded the discussion a bit more.
This theorem means that something that I wrote at Cauchy surface is not true; I fixed it in such a way as to link to globally hyperbolic (again) there.
Currently the entry says that global hyperbolicity of $X$ implies the existence of a homeomorphism $\mathbb{R} \times \Sigma \to X$ that exhibits the foliation by Cauchy surfaces.
I should check: can we not assume that this is a diffeomorphism with respect to the canonical smooth structure?
I don’t have more time for this right now. Maybe somebody knows directly? Otherwise I’ll try to check later.
added pointer to:
also pointer to:
Miguel Sánchez, Globally hyperbolic spacetimes: slicings, boundaries and counterexamples, Gen Relativ Gravit 54 124 (2022) [arXiv:2110.13672, doi:10.1007/s10714-022-03002-6]
Antonio N. Bernal, Miguel Sánchez, Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Commun. Math. Phys. 257 (2005) 43-50 [arXiv:gr-qc/0401112, doi:10.1007/s00220-005-1346-1]
Antonio N. Bernal, Miguel Sánchez, Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions, Lett. Math. Phys. 77 (2006) 183-197 [arXiv:gr-qc/0512095, doi:10.1007/s11005-006-0091-5]
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