Foliation

Builds on Smooth manifold Tangent bundle

Definition

Let $M$ be a smooth manifold of dimension $n$ , and let $p, q \ge 0$ be integers with $p + q = n$ . A foliation of dimension $p$ (and codimension $q$ ) of $M$ is a maximal atlas of charts $\varphi_i \colon U_i \to \R^p \times \R^q$ with the following property. The transition maps have the local form $\varphi_j \circ \varphi_i^{-1}(x, y) = \bigl(g_{ij}(x, y), h_{ij}(y)\bigr), \qquad x \in \R^p, \ y \in \R^q.$ In such a chart, the sets $\varphi_i^{-1}(\R^p \times \set{y}) \cap U_i$ are the plaques. Call two points of $M$ equivalent when they can be joined by a finite chain of plaques, each meeting the next. The leaves of the foliation are the equivalence classes of this relation.

Remarks

Each leaf is a connected injectively immersed submanifold of dimension $p$ , and the leaves partition $M$ . The tangent spaces of the leaves form a subbundle of the tangent bundle that is involutive; by the Frobenius theorem, foliations of dimension $p$ correspond to involutive rank $p$ subbundles.

Example: on the torus $\R^2 / \Z^2$ , the lines of slope $\alpha$ give a foliation of dimension $1$ . The charts are restrictions of the linear map $(s, t) \mapsto (s + \alpha t, t - \alpha s)$ composed with local inverses of the quotient map, and transition maps are translations, which have the required form. If $\alpha$ is irrational, each leaf is dense in the torus.

Non-example: on $\R^3$ with coordinates $(x, y, z)$ , the plane field spanned by $\partial_x$ and $\partial_y + x \, \partial_z$ is not involutive, because the bracket of these two fields is $\partial_z$ . So it is not the tangent field of any foliation by surfaces.

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Smooth manifold concept
Tangent bundle concept
Foliation definition

Tag	`S8TA91BAN8`
URI	`m2://S8TA91BAN8`
Kind	definition
Writing	W0 stub
MSC	57R30, 53C12
Defines	Foliation
Uses	Smooth manifold, Tangent bundle
Changed	2 November 2026

Foliation

Definition

Remarks

Reading path

Further reading

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