Foliation

In mathematics, a p-dimensional foliation is a partition of a manifold into submanifolds, all of the same dimension p, locally modeled on the decomposition of Rⁿ into the p-dimensional planes cut out by the equations ${\displaystyle x_{p+1}=a_{p+1},\ldots ,x_{n}=a_{n}}$. The submanifolds are called the leaves of the foliation.

The 3-sphere has a famous codimension-1 foliation called the Reeb foliation.

The submanifolds are required to be connected and injectively immersed, but they are not required to be embedded. For example, if m is a fixed irrational number, the torus ${\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}}$ is foliated by the set of straight lines in the torus of slope m. Each line is dense in the torus and is injectively immersed but not embedded.

If the manifold and 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.

The level sets of a smooth real-valued function on a manifold with no critical points form a codimension 1 foliation on the manifold. For example, in general relativity, spacetimes with some number of special dimensions and 1 time dimension are often foliated as the level sets of a smooth function whose gradient is timelike, so that the leaves are spacelike hypersurfaces. Every codimension 1 foliation locally arises this way, but generally does not arise this way globally. For example, 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.