Foliation

In order to give a more precise definition of foliation, it is necessary to define some auxiliary elements.

A 3-dimensional foliated chart with n = 3 and q = 1. The plaques are 2-dimensional and the transversals are 1-dimensional.

A rectangular neighborhood in Rⁿ is an open subset of the form B = J₁ × ⋅⋅⋅ × J_n, where J_i is a (possibly unbounded) relatively open interval in the ith coordinate axis. If J₁ is of the form (a,0], it is said that B has boundary[6]

${\displaystyle \partial B=\left\{\left(0,x^{2},\ldots ,x^{n}\right)\in B\right\}.}$

In the following definition, coordinate charts are considered that have values in R^p × R^q, allowing the possibility of manifolds with boundary and (convex) corners.

A foliated chart on the n-manifold M of codimension q is a pair (U,φ), where U ⊆ M is open and ${\displaystyle \varphi :U\to B_{\tau }\times B_{\pitchfork }}$ is a diffeomorphism, ${\displaystyle B_{\pitchfork }}$ being a rectangular neighborhood in R^q and ${\displaystyle B_{\tau }}$ a rectangular neighborhood in R^p. The set P_y = φ⁻¹(B_τ × {y}), where ${\displaystyle y\in B_{\pitchfork }}$, is called a plaque of this foliated chart. For each x ∈ B_τ, the set S_x = φ⁻¹({x} × ${\displaystyle B_{\pitchfork }}$) is called a transversal of the foliated chart. The set ∂_τU = φ⁻¹(B_τ × (∂${\displaystyle B_{\pitchfork }}$)) is called the tangential boundary of U and ${\displaystyle \partial _{\pitchfork }U}$ = φ⁻¹((∂B_τ) × ${\displaystyle B_{\pitchfork }}$) is called the transverse boundary of U.[7]

The foliated chart is the basic model for all foliations, the plaques being the leaves. The notation B_τ is read as "B-tangential" and ${\displaystyle B_{\pitchfork }}$ as "B-transverse". There are also various possibilities. If both ${\displaystyle B_{\pitchfork }}$ and B_τ have empty boundary, the foliated chart models codimension-q foliations of n-manifolds without boundary. If one, but not both of these rectangular neighborhoods has boundary, the foliated chart models the various possibilities for foliations of n-manifolds with boundary and without corners. Specifically, if ∂${\displaystyle B_{\pitchfork }}$ ≠ ∅ = ∂B_τ, then ∂U = ∂_τU is a union of plaques and the foliation by plaques is tangent to the boundary. If ∂B_τ ≠ ∅ = ∂${\displaystyle B_{\pitchfork }}$, then ∂U = ${\displaystyle \partial _{\pitchfork }U}$ is a union of transversals and the foliation is transverse to the boundary. Finally, if ∂${\displaystyle B_{\pitchfork }}$ ≠ ∅ ≠ ∂B_τ, this is a model of a foliated manifold with a corner separating the tangential boundary from the transverse boundary.[7]

(a) Foliation tangent to the boundary ∂${\displaystyle B_{\pitchfork }}$ ≠ ∅ = ∂B_τ; (b) Foliation transverse to the boundary ∂B_τ ≠ ∅ = ∂${\displaystyle B_{\pitchfork }}$; (c) Foliation with a corner separating the tangential boundary from the transverse boundary ∂${\displaystyle B_{\pitchfork }}$ ≠ ∅ ≠ ∂B_τ.

A foliated atlas of codimension q and class C^r (0 ≤ r ≤ ∞) on the n-manifold M is a C^r-atlas ${\displaystyle {\mathcal {U}}=\{(U_{\alpha },\varphi _{\alpha })\mid \alpha \in A\}}$ of foliated charts of codimension q which are coherently foliated in the sense that, whenever P and Q are plaques in distinct charts of ${\displaystyle {\mathcal {U}}}$, then P ∩ Q is open both in P and Q.[8]

A useful way to reformulate the notion of coherently foliated charts is to write for w ∈ U_α ∩ U_β [9]

${\displaystyle \varphi _{\alpha }(w)=\left(x_{\alpha }(w),y_{\alpha }(w)\right)\in B_{\tau }^{\alpha }\times B_{\pitchfork }^{\alpha },}$

${\displaystyle \varphi _{\beta }(w)=\left(x_{\beta }(w),y_{\beta }(w)\right)\in B_{\tau }^{\beta }\times B_{\pitchfork }^{\beta }.}$

The notation (U_α,φ_α) is often written (U_α,x_α,y_α), with [9]

${\displaystyle x_{\alpha }=\left(x_{\alpha }^{1},\dots ,x_{\alpha }^{p}\right),}$

${\displaystyle y_{\alpha }=\left(y_{\alpha }^{1},\dots ,y_{\alpha }^{q}\right).}$

On φ_β(U_α ∩ U_β) the coordinates formula can be changed as [9]

${\displaystyle g_{\alpha \beta }\left(x_{\beta },y_{\beta }\right)=\varphi _{\alpha }\circ \varphi _{\beta }^{-1}\left(x_{\beta },y_{\beta }\right)=\left(x_{\alpha }\left(x_{\beta },y_{\beta }\right),y_{\alpha }\left(x_{\beta },y_{\beta }\right)\right).}$

Plaques of U_α each meet two plaques of U_β.

The condition that (U_α,x_α,y_α) and (U_β,x_β,y_β) be coherently foliated means that, if P ⊂ U_α is a plaque, the connected components of P ∩ U_β lie in (possibly distinct) plaques of U_β. Equivalently, since the plaques of U_α and U_β are level sets of the transverse coordinates y_α and y_β, respectively, each point z ∈ U_α ∩ U_β has a neighborhood in which the formula

${\displaystyle y_{\alpha }=y_{\alpha }(x_{\beta },y_{\beta })=y_{\alpha }(y_{\beta })}$

is independent of x_β.[9]

The main use of foliated atlases is to link their overlapping plaques to form the leaves of a foliation. For this and other purposes, the general definition of foliated atlas above is a bit clumsy. One problem is that a plaque of (U_α,φ_α) can meet multiple plaques of (U_β,φ_β). It can even happen that a plaque of one chart meets infinitely many plaques of another chart. However, no generality is lost in assuming the situation to be much more regular as shown below.

Two foliated atlases ${\displaystyle {\mathcal {U}}}$ and ${\displaystyle {\mathcal {V}}}$ on M of the same codimension and smoothness class C^r are coherent ${\displaystyle \left({\mathcal {U}}\thickapprox {\mathcal {V}}\right)}$ if ${\displaystyle {\mathcal {U}}\cup {\mathcal {V}}}$ is a foliated C^r-atlas. Coherence of foliated atlases is an equivalence relation.[9]

Proof [9]

Reflexivity and symmetry are immediate. To prove transitivity let ${\displaystyle {\mathcal {U}}\thickapprox {\mathcal {V}}}$ and ${\displaystyle {\mathcal {V}}\thickapprox {\mathcal {W}}}$. Let (Uα,xα,yα) ∈ ${\displaystyle {\mathcal {U}}}$ and (Wλ,xλ,yλ) ∈ ${\displaystyle {\mathcal {W}}}$ and suppose that there is a point w ∈ Uα ∩ Wλ. Choose (Vδ,xδ,yδ) ∈ ${\displaystyle {\mathcal {V}}}$ such that w ∈ Vδ. By the above remarks, there is a neighborhood N of w in Uα ∩ Vδ ∩ Wλ such that ${\displaystyle y_{\delta }=y_{\delta }(y_{\lambda })\quad {\text{on}}\quad \varphi _{\lambda }(N),}$ ${\displaystyle y_{\alpha }=y_{\alpha }(y_{\delta })\quad {\text{on}}\quad \varphi _{\delta }(N),}$ and hence ${\displaystyle y_{\alpha }=y_{\alpha }\left(y_{\delta }(y_{\lambda })\right)\quad {\text{on}}\quad \varphi _{\delta }(N).}$ Since w ∈ Uα ∩ Wλ is arbitrary, it can be concluded that yα(xλ,yλ) is locally independent of xλ. It is thus proven that ${\displaystyle {\mathcal {U}}\thickapprox {\mathcal {W}}}$, hence that coherence is transitive.[10]

Sample charts in a regular foliated atlas.

Plaques and transversals defined above on open sets are also open. But one can speak also of closed plaques and transversals. Namely, if (U,φ) and (W,ψ) are foliated charts such that ${\displaystyle {\overline {U}}}$ (the closure of U) is a subset of W and φ = ψ|U then, if ${\displaystyle \varphi (U)=B_{\tau }\times B_{\pitchfork },}$ it can be seen that ${\displaystyle \psi |{\overline {U}}}$, written ${\displaystyle {\overline {\varphi }}}$, carries ${\displaystyle {\overline {U}}}$ diffeomorphically onto ${\displaystyle {\overline {B}}_{\tau }\times {\overline {B}}_{\pitchfork }.}$

A foliated atlas is said to be regular if

1. for each α ∈ A, ${\displaystyle {\overline {U}}_{\alpha }}$ is a compact subset of a foliated chart (W_α,ψ_α) and φ_α = ψ_α|U_α;
2. the cover {U_α | α ∈ A} is locally finite;
3. if (U_α,φ_α) and (U_β,φ_β) are elements of the foliated atlas, then the interior of each closed plaque P ⊂ ${\displaystyle {\overline {U}}_{\alpha }}$ meets at most one plaque in ${\displaystyle {\overline {U}}_{\beta }.}$ [11]

By property (1), the coordinates x_α and y_α extend to coordinates ${\displaystyle {\overline {x}}_{\alpha }}$ and ${\displaystyle {\overline {y}}_{\alpha }}$ on ${\displaystyle {\overline {U}}_{\alpha }}$ and one writes ${\displaystyle {\overline {\varphi }}_{\alpha }=\left({\overline {x}}_{\alpha },{\overline {y}}_{\alpha }\right).}$ Property (3) is equivalent to requiring that, if U_α ∩ U_β ≠ ∅, the transverse coordinate changes ${\displaystyle {\overline {y}}_{\alpha }={\overline {y}}_{\alpha }\left({\overline {x}}_{\beta },{\overline {y}}_{\beta }\right)}$ be independent of ${\displaystyle {\overline {x}}_{\beta }.}$ That is

${\displaystyle {\overline {g}}_{\alpha \beta }={\overline {\varphi }}_{\alpha }\circ {\overline {\varphi }}_{\beta }^{-1}:{\overline {\varphi }}_{\beta }\left({\overline {U}}_{\alpha }\cap {\overline {U}}_{\beta }\right)\rightarrow {\overline {\varphi }}_{\alpha }\left({\overline {U}}_{\alpha }\cap {\overline {U}}_{\beta }\right)}$

has the formula [11]

${\displaystyle {\overline {g}}_{\alpha \beta }\left({\overline {x}}_{\beta },{\overline {y}}_{\beta }\right)=\left({\overline {x}}_{\alpha }\left({\overline {x}}_{\beta },{\overline {y}}_{\beta }\right),{\overline {y}}_{\alpha }\left({\overline {y}}_{\beta }\right)\right).}$

Similar assertions hold also for open charts (without the overlines). The transverse coordinate map y_α can be viewed as a submersion

${\displaystyle y_{\alpha }:U_{\alpha }\rightarrow \mathbb {R} ^{q}}$

and the formulas y_α = y_α(y_β) can be viewed as diffeomorphisms

${\displaystyle \gamma _{\alpha \beta }:y_{\beta }\left(U_{\alpha }\cap U_{\beta }\right)\rightarrow y_{\alpha }\left(U_{\alpha }\cap U_{\beta }\right).}$

These satisfy the cocycle conditions. That is, on y_δ(U_α ∩ U_β ∩ U_δ),

${\displaystyle \gamma _{\alpha \delta }=\gamma _{\alpha \beta }\circ \gamma _{\beta \delta }}$

and, in particular,[12]

${\displaystyle \gamma _{\alpha \alpha }\equiv y_{\alpha }\left(U_{\alpha }\right),}$

${\displaystyle \gamma _{\alpha \beta }=\gamma _{\beta \alpha }^{-1}.}$

Using the above definitions for coherence and regularity it can be proven that every foliated atlas has a coherent refinement that is regular.[13]

Proof [13]

Fix a metric on M and a foliated atlas ${\displaystyle {\mathcal {W}}.}$ Passing to a subcover, if necessary, one can assume that ${\displaystyle {\mathcal {W}}=\left\{W_{j},\psi _{j}\right\}_{j=1}^{l}}$ is finite. Let ε > 0 be a Lebesgue number for ${\displaystyle {\mathcal {W}}.}$ That is, any subset X ⊆ M of diameter < ε lies entirely in some Wj. For each x ∈ M, choose j such that x ∈ Wj and choose a foliated chart (Ux, φx) such that x ∈ Ux ⊆ ${\displaystyle {\overline {U}}_{x}}$ ⊂ Wj, φx = ψj|Ux, diam(Ux) < ε/2. Suppose that Ux ⊂ Wk, k ≠ j, and write ψk = (xk,yk) as usual, where yk : Wk → Rq is the transverse coordinate map. This is a submersion having the plaques in Wk as level sets. Thus, yk restricts to a submersion yk : Ux → Rq. This is locally constant in xj; so choosing Ux smaller, if necessary, one can assume that yk|${\displaystyle {\overline {U}}_{x}}$ has the plaques of ${\displaystyle {\overline {U}}_{x}}$ as its level sets. That is, each plaque of Wk meets (hence contains) at most one (compact) plaque of ${\displaystyle {\overline {U}}_{x}}$. Since 1 < k < l < ∞, one can choose Ux so that, whenever Ux ⊂ Wk, distinct plaques of ${\displaystyle {\overline {U}}_{x}}$ lie in distinct plaques of Wk. Pass to a finite subatlas ${\displaystyle {\mathcal {U}}=\left\{U_{i},\varphi _{i}\right\}_{i=1}^{N}}$ of {(Ux,φx) | x ∈ M}. If Ui ∩ Uj ≠ 0, then diam(Ui ∪ Uj) < ε, and so there is an index k such that ${\displaystyle {\overline {U}}_{i}\cup {\overline {U}}_{j}\subseteq W_{k}.}$ Distinct plaques of ${\displaystyle {\overline {U}}_{i}}$ (respectively, of ${\displaystyle {\overline {U}}_{j}}$) lie in distinct plaques of Wk. Hence each plaque of ${\displaystyle {\overline {U}}_{i}}$ has interior meeting at most one plaque of ${\displaystyle {\overline {U}}_{j}}$ and vice versa. By construction, ${\displaystyle {\mathcal {U}}}$ is a coherent refinement of ${\displaystyle {\mathcal {W}}}$ and is a regular foliated atlas. If M is not compact, local compactness and second countability allows one to choose a sequence ${\displaystyle \left\{K_{i}\right\}_{i=0}^{\infty }}$ of compact subsets such that Ki ⊂ int Ki+1 for each i ≥ 0 and ${\displaystyle M=\bigcup _{i=1}^{\infty }K_{i}.}$ Passing to a subatlas, it is assumed that ${\displaystyle {\mathcal {W}}=\left\{W_{j},\psi _{j}\right\}_{j=0}^{\infty }}$ is countable and a strictly increasing sequence ${\displaystyle \left\{n_{l}\right\}_{l=0}^{\infty }}$ of positive integers can be found such that ${\displaystyle {\mathcal {W}}_{l}=\left\{W_{j},\psi _{j}\right\}_{j=0}^{n_{l}}}$ covers Kl. Let δl denote the distance from Kl to ∂Kl+1 and choose εl > 0 so small that εl < min{δl/2,εl-1} for l ≥ 1, ε0 < δ0/2, and εl is a Lebesgue number for ${\displaystyle {\mathcal {W}}_{l}}$ (as an open cover of Kl) and for ${\displaystyle {\mathcal {W}}_{l+1}}$ (as an open cover of Kl+1). More precisely, if X ⊂ M meets Kl (respectively, Kl+1) and diam X < εl, then X lies in some element of ${\displaystyle {\mathcal {W}}_{l}}$ (respectively, ${\displaystyle {\mathcal {W}}_{l+1}}$). For each x ∈ Kl ${\displaystyle \diagdown }$ int Kl-1, construct (Ux,φx) as for the compact case, requiring that ${\displaystyle {\overline {U}}_{x}}$ be a compact subset of Wj and that φx = ψj|Ux, some j ≤ nl. Also, require that diam${\displaystyle {\overline {U}}_{x}}$ < εl/2. As before, pass to a finite subcover ${\displaystyle \left\{U_{i},\varphi _{i}\right\}_{i=n_{l-1}+1}^{n_{l}}}$ of Kl ${\displaystyle \diagdown }$ int Kl-1. (Here, it is taken n−1 = 0.) This creates a regular foliated atlas ${\displaystyle {\mathcal {U}}=\left\{U_{i},\varphi _{i}\right\}_{i=1}^{\infty }}$ that refines ${\displaystyle {\mathcal {W}}}$ and is coherent with ${\displaystyle {\mathcal {W}}.}$.

Several alternative definitions of foliation exist depending on the way through which the foliation is achieved. The most common way to achieve a foliation is through decomposition reaching to the following

Decomposition through the coordinates function x : U→Rⁿ.

Definition. A p-dimensional, class C^r foliation of an n-dimensional manifold M is a decomposition of M into a union of disjoint connected submanifolds {L_α}_α∈A, called the leaves of the foliation, with the following property: Every point in M has a neighborhood U and a system of local, class C^r coordinates x=(x¹, ⋅⋅⋅, xⁿ) : U→Rⁿ such that for each leaf L_α, the components of U ∩ L_α are described by the equations x^p+1=constant, ⋅⋅⋅, xⁿ=constant. A foliation is denoted by ${\displaystyle {\mathcal {F}}}$={L_α}_α∈A.[5]

The notion of leaves allows for an intuitive way of thinking about a foliation. For a slightly more geometrical definition, p-dimensional foliation ${\displaystyle {\mathcal {F}}}$ of an n-manifold M may be thought of as simply a collection {M_a} of pairwise-disjoint, connected, immersed p-dimensional submanifolds (the leaves of the foliation) of M, such that for every point x in M, there is a chart ${\displaystyle (U,\varphi )}$ with U homeomorphic to Rⁿ containing x such that every leaf, M_a, meets U in either the empty set or a countable collection of subspaces whose images under ${\displaystyle \varphi }$ in ${\displaystyle \varphi (M_{a}\cap U)}$ are p-dimensional affine subspaces whose first n − p coordinates are constant.

Locally, every foliation is a submersion allowing the following

Definition. Let M and Q be manifolds of dimension n and q≤n respectively, and let f : M→Q be a submersion, that is, suppose that the rank of the function differential (the Jacobian) is q. It follows from the Implicit Function Theorem that ƒ induces a codimension-q foliation on M where the leaves are defined to be the components of f⁻¹(x) for x ∈ Q.[5]

This definition describes a dimension-p foliation ${\displaystyle {\mathcal {F}}}$ of an n-dimensional manifold M that is a covered by charts U_i together with maps

${\displaystyle \varphi _{i}:U_{i}\to \mathbb {R} ^{n}}$

such that for overlapping pairs U_i, U_j the transition functions φ_ij : Rⁿ → Rⁿ defined by

${\displaystyle \varphi _{ij}=\varphi _{j}\varphi _{i}^{-1}}$

take the form

${\displaystyle \varphi _{ij}(x,y)=(\varphi _{ij}^{1}(x),\varphi _{ij}^{2}(x,y))}$

where x denotes the first q = n − p coordinates, and y denotes the last p co-ordinates. That is,

${\displaystyle {\begin{aligned}\varphi _{ij}^{1}:{}&\mathbb {R} ^{q}\to \mathbb {R} ^{q}\\\varphi _{ij}^{2}:{}&\mathbb {R} ^{n}\to \mathbb {R} ^{p}\end{aligned}}}$

The splitting of the transition functions φ_ij into ${\displaystyle \varphi _{ij}^{1}(x)}$ and ${\displaystyle \varphi _{ij}^{2}(x,y)}$ as a part of the submersion is completely analogous to the splitting of ${\displaystyle {\overline {g}}_{\alpha \beta }}$ into ${\displaystyle {\overline {y}}_{\alpha }\left({\overline {y}}_{\beta }\right)}$ and ${\displaystyle {\overline {x}}_{\alpha }\left({\overline {x}}_{\beta },{\overline {y}}_{\beta }\right)}$ as a part of the definition of a regular foliated atlas. This makes possible another definition of foliations in terms of regular foliated atlases. To this end, one has to prove first that every regular foliated atlas of codimension q is associated to a unique foliation ${\displaystyle {\mathcal {F}}}$ of codimension q.[13]

Proof [13]

Let ${\displaystyle {\mathcal {U}}=\left\{U_{a},\varphi _{\alpha }\right\}_{\alpha \in A}}$ be a regular foliated atlas of codimension q. Define an equivalence relation on M by setting x ~ y if and only if either there is a ${\displaystyle {\mathcal {U}}}$-plaque P0 such that x,y ∈ P0 or there is a sequence L = {P0,P1,⋅⋅⋅,Pp} of ${\displaystyle {\mathcal {U}}}$-plaques such that x ∈ P0, y ∈ Pp, and Pi ∩ Pi-1 ≠ ∅ with 1 ≤ i ≤ p. The sequence L will be called a plaque chain of length p connecting x and y. In the case that x,y ∈ P0, it is said that {P0} is a plaque chain of length 0 connecting x and y. The fact that ~ is an equivalence relation is clear. It is also clear that each equivalence class L is a union of plaques. Since ${\displaystyle {\mathcal {U}}}$-plaques can only overlap in open subsets of each other, L is locally a topologically immersed submanifold of dimension n − q. The open subsets of the plaques P ⊂ L form the base of a locally Euclidean topology on L of dimension n − q and L is clearly connected in this topology. It is also trivial to check that L is Hausdorff. The main problem is to show that L is second countable. Since each plaque is 2nd countable, the same will hold for L if it is shown that the set of ${\displaystyle {\mathcal {U}}}$-plaques in L is at most countably infinite. Fix one such plaque P0. By the definition of a regular, foliated atlas, P0 meets only finitely many other plaques. That is, there are only finitely many plaque chains {P0,Pi} of length 1. By induction on the length p of plaque chains that begin at P0, it is similarly proven that there are only finitely many of length ≤ p. Since every ${\displaystyle {\mathcal {U}}}$-plaque in L is, by the definition of ~, reached by a finite plaque chain starting at P0, the assertion follows.

As shown in the proof, the leaves of the foliation are equivalence classes of plaque chains of length ≤ p which are also topologically immersed Hausdorff p-dimensional submanifolds. Next, it is shown that the equivalence relation of plaques on a leaf is expressed in equivalence of coherent foliated atlases in respect to their association with a foliation. More specifically, if ${\displaystyle {\mathcal {U}}}$ and ${\displaystyle {\mathcal {V}}}$ are foliated atlases on M and if ${\displaystyle {\mathcal {U}}}$ is associated to a foliation ${\displaystyle {\mathcal {F}}}$ then ${\displaystyle {\mathcal {U}}}$ and ${\displaystyle {\mathcal {V}}}$ are coherent if and only if ${\displaystyle {\mathcal {V}}}$ is also associated to ${\displaystyle {\mathcal {F}}}$.[10]

Proof [10]

If ${\displaystyle {\mathcal {V}}}$ is also associated to ${\displaystyle {\mathcal {F}}}$, every leaf L is a union of ${\displaystyle {\mathcal {V}}}$-plaques and of ${\displaystyle {\mathcal {U}}}$-plaques. These plaques are open subsets in the manifold topology of L, hence intersect in open subsets of each other. Since plaques are connected, a ${\displaystyle {\mathcal {U}}}$-plaque cannot intersect a ${\displaystyle {\mathcal {V}}}$-plaque unless they lie in a common leaf; so the foliated atlases are coherent. Conversely, if we only know that ${\displaystyle {\mathcal {U}}}$ is associated to ${\displaystyle {\mathcal {F}}}$ and that ${\displaystyle {\mathcal {V}}\approx {\mathcal {U}}}$, let Q be a ${\displaystyle {\mathcal {V}}}$-plaque. If L is a leaf of ${\displaystyle {\mathcal {F}}}$ and w ∈ L ∩ Q, let P ∈ L be a ${\displaystyle {\mathcal {U}}}$-plaque with w ∈ P. Then P ∩ Q is an open neighborhood of w in Q and P ∩ Q ⊂ L ∩ Q. Since w ∈ L ∩ Q is arbitrary, it follows that L ∩ Q is open in Q. Since L is an arbitrary leaf, it follows that Q decomposes into disjoint open subsets, each of which is the intersection of Q with some leaf of ${\displaystyle {\mathcal {F}}}$. Since Q is connected, L ∩ Q = Q. Finally, Q is an arbitrary ${\displaystyle {\mathcal {V}}}$-plaque, and so ${\displaystyle {\mathcal {V}}}$ is associated to ${\displaystyle {\mathcal {F}}}$.

It is now obvious that the correspondence between foliations on M and their associated foliated atlases induces a one-to-one correspondence between the set of foliations on M and the set of coherence classes of foliated atlases or, in other words, a foliation ${\displaystyle {\mathcal {F}}}$ of codimension q and class C^r on M is a coherence class of foliated atlases of codimension q and class C^r on M.[14] By Zorn's lemma, it is obvious that every coherence class of foliated atlases contains a unique maximal foliated atlas. Thus,

Definition. A foliation of codimension q and class C^r on M is a maximal foliated C^r-atlas of codimension q on M.[14]

In practice, a relatively small foliated atlas is generally used to represent a foliation. Usually, it is also required this atlas to be regular.

In the chart U_i, the stripes x = constant match up with the stripes on other charts U_j. These submanifolds piece together from chart to chart to form maximal connected injectively immersed submanifolds called the leaves of the foliation.

If one shrinks the chart U_i it can be written as U_ix × U_iy, where U_ix ⊂ R^n−p, U_iy ⊂ R^p, U_iy is homeomorphic to the plaques, and the points of U_ix parametrize the plaques in U_i. If one picks y₀ in U_iy, then U_ix × {y₀} is a submanifold of U_i that intersects every plaque exactly once. This is called a local transversal section of the foliation. Note that due to monodromy global transversal sections of the foliation might not exist.

The case r = 0 is rather special. Those C⁰ foliations that arise in practice are usually "smooth-leaved". More precisely, they are of class C^r,0, in the following sense.

Definition. A foliation ${\displaystyle {\mathcal {F}}}$ is of class C^r,k, r > k ≥ 0, if the corresponding coherence class of foliated atlases contains a regular foliated atlas {U_α,x_α,y_α}_α∈A such that the change of coordinate formula

${\displaystyle g_{\alpha \beta }(x_{\beta },y_{\beta })=(x_{\alpha }(x_{\beta },y_{\beta }),y_{\alpha }(y_{\beta })).}$

is of class C^k, but x_α is of class C^r in the coordinates x_β and its mixed x_β partials of orders ≤ r are C^k in the coordinates (x_β,y_β).[14]

The above definition suggests the more general concept of a foliated space or abstract lamination. One relaxes the condition that the transversals be open, relatively compact subsets of R^q, allowing the transverse coordinates y_α to take their values in some more general topological space Z. The plaques are still open, relatively compact subsets of R^p, the change of transverse coordinate formula y_α(y_β) is continuous and x_α(x_β,y_β) is of class C^r in the coordinates x_β and its mixed x_β partials of orders ≤ r are continuous in the coordinates (x_β,y_β). One usually requires M and Z to be locally compact, second countable and metrizable. This may seem like a rather wild generalization, but there are contexts in which it is useful.[15]

Let (M, ${\displaystyle {\mathcal {F}}}$) be a foliated manifold. If L is a leaf of ${\displaystyle {\mathcal {F}}}$ and s is a path in L, one is interested in the behavior of the foliation in a neighborhood of s in M. Intuitively, an inhabitant of the leaf walks along the path s, keeping an eye on all of the nearby leaves. As they (hereafter denoted by s(t)) proceed, some of these leaves may "peel away", getting out of visual range, others may suddenly come into range and approach L asymptotically, others may follow along in a more or less parallel fashion or wind around L laterally, etc. If s is a loop, then s(t) repeatedly returns to the same point s(t₀) as t goes to infinity and each time more and more leaves may have spiraled into view or out of view, etc. This behavior, when appropriately formalized, is called the holonomy of the foliation.

Holonomy is implemented on foliated manifolds in various specific ways: the total holonomy group of foliated bundles, the holonomy pseudogroup of general foliated manifolds, the germinal holonomy groupoid of general foliated manifolds, the germinal holonomy group of a leaf, and the infinitesimal holonomy group of a leaf.

Foliated bundles

The easiest case of holonomy to understand is the total holonomy of a foliated bundle. This is a generalization of the notion of a Poincaré map.

A cross section N and first return map f where M = S¹ × D² and N = D².

The term "first return (recurrence) map" comes from the theory of dynamical systems. Let Φ_t be a nonsingular C^r flow (r ≥ 1) on the compact n-manifold M. In applications, one can imagine that M is a cyclotron or some closed loop with fluid flow. If M has a boundary, the flow is assumed to be tangent to the boundary. The flow generates a 1-dimensional foliation ${\displaystyle {\mathcal {F}}}$. If one remembers the positive direction of flow, but otherwise forgets the parametrization (shape of trajectory, velocity, etc.), the underlying foliation ${\displaystyle {\mathcal {F}}}$ is said to be oriented. Suppose that the flow admits a global cross section N. That is, N is a compact, properly embedded, C^r submanifold of M of dimension n – 1, the foliation ${\displaystyle {\mathcal {F}}}$ is transverse to N, and every flow line meets N. Because the dimensions of N and of the leaves are complementary, the transversality condition is that

${\displaystyle T_{y}(M)=T_{y}({\mathcal {F}})\oplus T_{y}(N){\text{ for each }}y\in N.}$

Let y ∈ N and consider the ω-limit set ω(y) of all accumulation points in M of all sequences ${\displaystyle \left\{\Phi _{t_{k}}(y)\right\}_{k=1}^{\infty }}$, where t_k goes to infinity. It can be shown that ω(y) is compact, nonempty, and a union of flow lines. If ${\displaystyle z=\lim _{k\rightarrow \infty }\Phi _{t_{k}}\in \omega (y),}$ there is a value t* ∈ R such that Φ_t*(z) ∈ N and it follows that

${\displaystyle \lim _{k\to \infty }\Phi _{t_{k}+t^{\ast }}(y)=\Phi _{t^{\ast }}(z)\in N.}$

Since N is compact and ${\displaystyle {\mathcal {F}}}$ is transverse to N, it follows that the set {t > 0 | Φ_t(y) ∈ N} is a monotonically increasing sequence ${\displaystyle \{\tau _{k}(y)\}_{k=1}^{\infty }}$ that diverges to infinity.

As y ∈ N varies, let τ(y) = τ₁(y), defining in this way a positive function τ ∈ C^r(N) (the first return time) such that, for arbitrary y ∈ N, Φ_t(y) ∉ N, 0 < t < τ(y), and Φ_τ(y)(y) ∈ N.

Define f : N → N by the formula f(y) = Φ_τ(y)(y). This is a C^r map. If the flow is reversed, exactly the same construction provides the inverse f⁻¹; so f ∈ Diff^r(N). This diffeomorphism is the first return map and τ is called the first return time. While the first return time depends on the parametrization of the flow, it should be evident that f depends only on the oriented foliation ${\displaystyle {\mathcal {F}}}$. It is possible to reparametrize the flow Φ_t, keeping it nonsingular, of class C^r, and not reversing its direction, so that τ ≡ 1.

The assumption that there is a cross section N to the flow is very restrictive, implying that M is the total space of a fiber bundle over S¹. Indeed, on R × N, define ~_f to be the equivalence relation generated by

${\displaystyle (t,y)\sim _{f}(t-1,f(y)).}$

Equivalently, this is the orbit equivalence for the action of the additive group Z on R × N defined by

${\displaystyle k\cdot (t,y)=(t-k,f^{k}(y)),}$

for each k ∈ Z and for each (t,y) ∈ R × N. The mapping cylinder of f is defined to be the C^r manifold

${\displaystyle M_{f}=(\mathbb {R} \times N)/{\sim _{f}}.}$

By the definition of the first return map f and the assumption that the first return time is τ ≡ 1, it is immediate that the map

${\displaystyle \Phi$ :\mathbb {R} \times N\rightarrow M.}

defined by the flow, induces a canonical C^r diffeomorphism

${\displaystyle \varphi :M_{f}\rightarrow M.}$

If we make the identification M_f = M, then the projection of R × N onto R induces a C^r map

${\displaystyle \pi :M\rightarrow \mathbb {R} /\mathbb {Z} =S^{1}}$

that makes M into the total space of a fiber bundle over the circle. This is just the projection of S¹ × D² onto S¹. The foliation ${\displaystyle {\mathcal {F}}}$ is transverse to the fibers of this bundle and the bundle projection π, restricted to each leaf L, is a covering map π : L → S¹. This is called a foliated bundle.

Take as basepoint x₀ ∈ S¹ the equivalence class 0 + Z; so π⁻¹(x₀) is the original cross section N. For each loop s on S¹, based at x₀, the homotopy class [s] ∈ π₁(S¹,x₀) is uniquely characterized by deg s ∈ Z. The loop s lifts to a path in each flow line and it should be clear that the lift s_y that starts at y ∈ N ends at f^k(y) ∈ N, where k = deg s. The diffeomorphism f^k ∈ Diff^r(N) is also denoted by h_s and is called the total holonomy of the loop s. Since this depends only on [s], this is a definition of a homomorphism

${\displaystyle h:\pi _{1}(S^{1},x_{0})\rightarrow \operatorname {Diff} ^{\,r}(N),}$

called the total holonomy homomorphism for the foliated bundle.

Using fiber bundles in a more direct manner, let (M,${\displaystyle {\mathcal {F}}}$) be a foliated n-manifold of codimension q. Let π : M → B be a fiber bundle with q-dimensional fiber F and connected base space B. Assume that all of these structures are of class C^r, 0 ≤ r ≤ ∞, with the condition that, if r = 0, B supports a C¹ structure. Since every maximal C¹ atlas on B contains a C^∞ subatlas, no generality is lost in assuming that B is as smooth as desired. Finally, for each x ∈ B, assume that there is a connected, open neighborhood U ⊆ B of x and a local trivialization

${\displaystyle {\begin{matrix}\pi ^{-1}(U)&{\xrightarrow {\varphi }}&U\times {F}\\\scriptstyle {\pi }{\Bigg \downarrow }&{\qquad }&{\Bigg \downarrow }{\scriptstyle {p}}\\U&{\xrightarrow {\text{id}}}&U\end{matrix}}}$

where φ is a C^r diffeomorphism (a homeomorphism, if r = 0) that carries ${\textstyle {\mathcal {F}}\mid \pi ^{-1}(U)}$ to the product foliation {U × {y}}_y ∈ F. Here, ${\textstyle {\mathcal {F}}\mid \pi ^{-1}(U)}$ is the foliation with leaves the connected components of L ∩ π⁻¹(U), where L ranges over the leaves of ${\displaystyle {\mathcal {F}}}$. This is the general definition of the term "foliated bundle" (M,${\displaystyle {\mathcal {F}}}$,π) of class C^r.

${\displaystyle {\mathcal {F}}}$ is transverse to the fibers of π (it is said that ${\displaystyle {\mathcal {F}}}$ is transverse to the fibration) and that the restriction of π to each leaf L of ${\displaystyle {\mathcal {F}}}$ is a covering map π : L → B. In particular, each fiber F_x = π⁻¹(x) meets every leaf of ${\displaystyle {\mathcal {F}}}$. The fiber is a cross section of ${\displaystyle {\mathcal {F}}}$ in complete analogy with the notion of a cross section of a flow.

The foliation ${\displaystyle {\mathcal {F}}}$ being transverse to the fibers does not, of itself, guarantee that the leaves are covering spaces of B. A simple version of the problem is a foliation of R², transverse to the fibration

${\displaystyle \pi$ :\mathbb {R} ^{2}\rightarrow \mathbb {R} ,}

${\displaystyle \pi (x,y)=x,}$

but with infinitely many leaves missing the y-axis. In the respective figure, it is intended that the "arrowed" leaves, and all above them, are asymptotic to the axis x = 0. One calls such a foliation incomplete relative to the fibration, meaning that some of the leaves "run off to infinity" as the parameter x ∈ B approaches some x₀ ∈ B. More precisely, there may be a leaf L and a continuous path s : [0,a) → L such that lim_t→a−π(s(t)) = x₀ ∈ B, but lim_t→a−s(t) does not exist in the manifold topology of L. This is analogous to the case of incomplete flows, where some flow lines "go to infinity" in finite time. Although such a leaf L may elsewhere meet π⁻¹(x₀), it cannot evenly cover a neighborhood of x₀, hence cannot be a covering space of B under π. When F is compact, however, it is true that transversality of ${\displaystyle {\mathcal {F}}}$ to the fibration does guarantee completeness, hence that ${\textstyle (M,{\mathcal {F}},\pi )}$ is a foliated bundle.

There is an atlas ${\displaystyle {\mathcal {U}}}$ = {U_α,x_α}_α∈A on B, consisting of open, connected coordinate charts, together with trivializations φ_α : π⁻¹(U_α) → U_α × F that carry ${\displaystyle {\mathcal {F}}}$|π⁻¹(U_α) to the product foliation. Set W_α = π⁻¹(U_α) and write φ_α = (x_α,y_α) where (by abuse of notation) x_α represents x_α ∘ π and y_α : π⁻¹(U_α) → F is the submersion obtained by composing φ_α with the canonical projection U_α × F → F.

The atlas ${\displaystyle {\mathcal {W}}}$ = {W_α,x_α,y_α}_α∈A plays a role analogous to that of a foliated atlas. The plaques of W_α are the level sets of y_α and this family of plaques is identical to F via y_α. Since B is assumed to support a C^∞ structure, according to the Whitehead theorem one can fix a Riemannian metric on B and choose the atlas ${\displaystyle {\mathcal {U}}}$ to be geodesically convex. Thus, U_α ∩ U_β is always connected. If this intersection is nonempty, each plaque of W_α meets exactly one plaque of W_β. Then define a holonomy cocycle ${\displaystyle \gamma =\left\{\gamma _{\alpha \beta }\right\}_{\alpha ,\beta \in A}}$ by setting

${\displaystyle \gamma _{\alpha \beta }=y_{\alpha }\circ y_{\beta }^{-1}:F\rightarrow F.}$

There is a close relationship, assuming everything is smooth, with vector fields: given a vector field X on M that is never zero, its integral curves will give a 1-dimensional foliation. (i.e. a codimension n − 1 foliation).

This observation generalises to the Frobenius theorem, saying that the necessary and sufficient conditions for a distribution (i.e. an n − p dimensional subbundle of the tangent bundle of a manifold) to be tangent to the leaves of a foliation, is that the set of vector fields tangent to the distribution are closed under Lie bracket. One can also phrase this differently, as a question of reduction of the structure group of the tangent bundle from GL(n) to a reducible subgroup.

The conditions in the Frobenius theorem appear as integrability conditions; and the assertion is that if those are fulfilled the reduction can take place because local transition functions with the required block structure exist. For example, in the codimension 1 case, we can define the tangent bundle of the foliation as ker(α), for some (non-canonical) α ∈ Ω¹ (i.e. a non-zero co-vector field). A given α is integrable iff α ∧ dα = 0 everywhere.

There is a global foliation theory, because topological constraints exist. For example, in the surface case, an everywhere non-zero vector field can exist on an orientable compact surface only for the torus. This is a consequence of the Poincaré–Hopf index theorem, which shows the Euler characteristic will have to be 0. There are many deep connections with contact topology, which is the "opposite" concept, requiring that the integrability condition is never satisfied.

Haefliger (1970) gave a necessary and sufficient condition for a distribution on a connected non-compact manifold to be homotopic to an integrable distribution. Thurston (1974, 1976) showed that any compact manifold with a distribution has a foliation of the same dimension.

• G-structure – Structure group sub-bundle on a tangent frame bundlePages displaying short descriptions of redirect targets
• Haefliger structure – Generalization of a foliation closed under taking pullbacks.
• Lamination – Partitioned topological space
• Reeb foliation – particular foliation of the 3-sphere introduced by the French mathematician Georges ReebPages displaying wikidata descriptions as a fallback.
• Taut foliation

• Foliations at the Manifold Atlas