  The splitting of the tangent bundle of space-time into temporal and spatial
sub-bundles defines an almost-product structure. In particular, any space-time
metric can be locally expressed in time-orthogonal form, in such a way that
whether or not that almost-product structure is locally generated by a
coordinate chart is a matter of the integrability of the Pfaff equation that
the temporal 1-form of that expression for the metric defines. When one applies
that analysis to the known exact solutions to the Einstein field equations, one
finds that many of the common ones are completely-integrable, although some of
the physically-interesting ones are not.
