  Under normal circumstances most members of the general relativity community
focus almost exclusively on the local properties of spacetime, such as the
locally Euclidean structure of the manifold and the Lorentzian signature of the
metric tensor. When combined with the classical Einstein field equations this
gives an extremely successful empirical model of classical gravity and
classical matter --- at least as long as one does not ask too many awkward
questions about global issues, (such as global topology and global causal
structure). We feel however that this is a tactical error --- even without
invoking full-fledged "quantum gravity" we know that the standard model of
particle physics is also an extremely good representation of some parts of
empirical reality; and we had better be able to carry over all the good
features of the standard model of particle physics --- at least into the realm
of semi-classical quantum gravity. Doing so gives us some interesting global
features that spacetime should possess: On physical grounds spacetime should be
space-orientable, time-orientable, and spacetime-orientable, and it should
possess a globally defined tetrad (vierbein, or in general a globally defined
vielbein/n-bein). So on physical grounds spacetime should be parallelizable.
This strongly suggests that the metric is not the fundamental physical
quantity; a very good case can be made for the tetrad being more fundamental
than the metric. Furthermore, a globally-defined "almost complex structure" is
almost unavoidable. Ideas along these lines have previously been mooted, but
much is buried in the pre-arXiv literature and is either forgotten or
inaccessible. We shall revisit these ideas taking a perspective very much based
on empirical physical observation.
