It seems to me there should be a statement of the theroem involving language that I find more intuitive, but I haven't seen any that really speak to me. What's more intuitive to me are Integral curves of vector fields, and how they wind up describing time-like congruences.wiki said:In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, the theorem gives necessary and sufficient conditions for the existence of a foliation by maximal integral manifolds each of whose tangent bundles are spanned by a given family of vector fields (satisfying an integrability condition) in much the same way as an integral curve may be assigned to a single vector field. The theorem is foundational in differential topology and calculus on manifolds.
For hypersurfaces, and for the conditions when two spacetimes are joined along a hypersurface, you might want to look at the first three chapters or so of the advanced, but somewhat pedagogical, "A Relativist's Toolkit: The Mathematics of Black-Mechanics" by Eric Poisson,Can anyone recommend me good books on the topics of Spacetime Hypersurfaces and Foliations of Space Time?