foliation

Foliation is a geometric structure on a manifold. In differential geometry, a foliation of a smooth manifold M of dimension n is a partition of M into connected immersed submanifolds called leaves, all of the same dimension p, where 0 < p < n. The collection locally looks like a product: every point has a neighborhood U equipped with a chart φ: U → R^p × R^q (with p+q=n) such that each plaque R^p × {y} lies in a single leaf. Equivalently, a foliation can be described by an integrable subbundle F of the tangent bundle TM of rank p; Frobenius’ theorem says F integrates to a foliation precisely when the space of sections of F is closed under the Lie bracket.

Examples include the trivial foliation of M by itself (p = n), the product foliation on a product manifold M × N by slices M × {y}, and foliations defined by level sets of a submersion f: U → R^q (with leaves of dimension p = n − q).

Foliations are studied for their geometric and dynamical properties. Leaves can be compact or noncompact, and they may be dense or form a regular foliation with a nice leaf space. The theory uses concepts such as holonomy, transverse structures, and the holonomy groupoid to capture how leaves fit together.

Geology and materials science use the term foliation to describe planar layering or fabric within rocks or minerals, produced by directed pressure during metamorphism. Common manifestations include slaty cleavage and schistosity, which reflect the preferred orientation of platy or elongated minerals and influence mechanical properties.