Transverse geometry and physical observers

It is proposed that the mathematical formalism that is most appropriate for the study of spatially non-integrable cosmological models is the transverse geometry of a one-dimensional foliation (congruence) defined by a physical observer. By that means, one can discuss the geometry of space, as viewed by that observer, without the necessity of introducing a complementary sub-bundle to the line bundle of the observer or a codimension-one foliation transverse to the foliation of the observer. The concept of groups of transverse isometries acting on such a spacetime and the relationship of transverse geometry to spacetime threadings (1+3 decompositions) is also discussed.