On the local integrability of almost-product structures defined by space-time metrics

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.