Foliations

A foliation on a smooth manifold M is a geometric structure that partitions M into connected immersed submanifolds called leaves, all of which have the same dimension p. The complementary dimension q = n − p is the codimension of the foliations. A foliation can be described by an atlas of foliated charts (U, φ) with φ: U → R^p × R^q such that the plaques φ^{-1}(R^p × {y}) are pieces of leaves, and the changes of charts preserve this product structure. Equivalently, locally M looks like a product of a leaf and a transverse space.

Equivalently, a foliation determines a rank-p subbundle TF of the tangent bundle TM, called the tangent distribution, which is involutive (closed under the Lie bracket). By Frobenius’ theorem, TF integrates to leaves, and each leaf is an immersed p-dimensional submanifold tangent to TF at every point. The leaf through a point is the maximal connected integral manifold of TF.

Examples include the level sets of a submersion f: M → N, which form a foliation by dimension n − dim N; the product foliation on N × F given by leaves N × {x}; the Reeb foliation of the 3-sphere, a nontrivial foliation with both compact and noncompact leaves; and the irrational slope foliation of the torus T^2, where leaves are dense lines of irrational slope.

Key features include that the space of leaves M/F can be highly non-Hausdorff; holonomy describes how nearby leaves twist around each other. Transverse geometry and basic cohomology study the structure perpendicular to the leaves. Foliation theory intersects dynamics, geometry, and topology, with refinements such as Riemannian foliations, minimal foliations, and characteristic classes like the Godbillon–Vey class.