Foliation

In mathematics (differential geometry), a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rⁿ into the cosets x + R^p of the standardly embedded subspace R^p. The equivalence classes are called the leaves of the foliation.^[1] If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class C^r), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class C^r it is usually understood that r ≥ 1 (otherwise, C⁰ is a topological foliation).^[2] The number p (the dimension of the leaves) is called the dimension of the foliation and q = n − p is called its codimension.

In some papers on general relativity by mathematical physicists, the term foliation (or slicing) is used to describe a situation where the relevant Lorentz manifold (a (p+1)-dimensional spacetime) has been decomposed into hypersurfaces of dimension p, specified as the level sets of a real-valued smooth function (scalar field) whose gradient is everywhere non-zero; this smooth function is moreover usually assumed to be a time function, meaning that its gradient is everywhere time-like, so that its level-sets are all space-like hypersurfaces. In deference to standard mathematical terminology, these hypersurface are often called the leaves (or sometimes slices) of the foliation.^[3] Note that while this situation does constitute a codimension-1 foliation in the standard mathematical sense, examples of this type are actually globally trivial; while the leaves of a (mathematical) codimension-1 foliation are always locally the level sets of a function, they generally cannot be expressed this way globally,^[4][5] as a leaf may pass through a local-trivializing chart infinitely many times, and the holonomy around a leaf may also obstruct the existence of a globally-consistent defining functions for the leaves. For example, while the 3-sphere has a famous codimension-1 foliation discovered by Reeb, a codimension-1 foliation of a closed manifold cannot be given by the level sets of a smooth function, since a smooth function on a closed manifold necessarily has critical points at its maxima and minima.