Foliaation

Foliaation is most commonly spelled foliation and refers to a concept in differential geometry and topology that describes a way to partition a smooth manifold into disjoint submanifolds called leaves, which fit together smoothly. If M is a smooth manifold of dimension n, a foliation of codimension q (so leaves have dimension p = n − q) decomposes M into connected immersed p-dimensional submanifolds that locally look like a product. Equivalently, there exists an atlas of charts such that within each chart the leaves correspond to the slices through the chart, and the transition maps preserve this leaf structure.

Formally, a foliation can be described by an integrable tangent distribution E ⊂ TM of rank p: at every point x ∈ M, E_x is a p-dimensional subspace of the tangent space T_xM, and E is involutive in the sense of Frobenius. The Frobenius theorem guarantees that E is tangent to an actual p-dimensional submanifold through each point, and these submanifolds piece together to form the leaves of the foliation. Conversely, a foliation determines a smooth, integrable distribution.

Common examples include the product foliation of a product manifold M × N by the slices M × {y}, and the foliation of R^n by parallel planes z = constant, or more generally by level sets of a submersion with regular values. More intricate examples arise in three-dimensional geometry, such as the Reeb foliation of the solid torus.

Key concepts related to foliations include holonomy, which describes how nearby leaves twist around each other, and invariants like the Godbillon–Vey class in codimension-1 foliations. Foliations connect to dynamical systems, geometry, and topology, providing a framework to study how local integrability leads to global structural patterns.