Inmathematics (differential geometry), afoliation is anequivalence relation on ann-manifold, theequivalence classes being connected,injectivelyimmersed submanifolds, all of the same dimensionp, modeled on thedecomposition of thereal coordinate spaceRⁿ into thecosetsx +R^p of the standardlyembeddedsubspaceR^p. The equivalence classes are called theleaves of the foliation.^[1] If the manifold and/or the submanifolds are required to have apiecewise-linear,differentiable (of classC^r), oranalytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of classC^r it is usually understood thatr ≥ 1 (otherwise,C⁰ is a topological foliation).^[2] The numberp (the dimension of the leaves) is called the dimension of the foliation andq =n −p is called itscodimension.

In some papers ongeneral relativity by mathematical physicists, the term foliation (orslicing) is used to describe a situation where the relevantLorentz manifold (a (p+1)-dimensionalspacetime) has been decomposed intohypersurfaces of dimensionp, specified as the level sets of a real-valuedsmooth function (scalar field) whosegradient is everywhere non-zero; this smooth function is moreover usually assumed to be atime function, meaning that its gradient is everywheretime-like, so that its level-sets are all space-like hypersurfaces. In deference to standard mathematical terminology, these hypersurface are often called the leaves (or sometimesslices) of the foliation.^[3] Note that while this situation does constitute a codimension-1 foliation in the standard mathematical sense, examples of this type are actually globally trivial; while the leaves of a (mathematical) codimension-1 foliation are alwayslocally the level sets of a function, they generally cannot be expressed this way globally,^[4][5] as a leaf may pass through a local-trivializing chart infinitely many times, and the holonomy around a leaf may also obstruct the existence of a globally-consistent defining functions for the leaves. For example, while the3-sphere has a famous codimension-1 foliation discovered by Reeb, a codimension-1 foliation of a closed manifold cannot be given by the level sets of a smooth function, since a smooth function on a closed manifold necessarily has critical points at its maxima and minima.

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

Arectangularneighborhood inRⁿ is anopensubset of the formB =J₁ × ⋅⋅⋅ ×J_n, whereJ_i is a (possibly unbounded) relatively open interval in theith coordinate axis. IfJ₁ is of the form (a,0], it is said thatB hasboundary^[6]

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

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

Afoliated chart on then-manifoldM of codimensionq is a pair (U,φ), whereU ⊆M is open and${\displaystyle \phi :U\to B_{\tau }\times B_{⋔}}$ is adiffeomorphism,${\displaystyle B_{⋔}}$ being a rectangular neighborhood inR^q and${\displaystyle B_{\tau }}$ a rectangular neighborhood inR^p. The setP_y =φ⁻¹(B_τ × {y}), where${\displaystyle y\in B_{⋔}}$, is called aplaque of this foliated chart. For each x ∈B_τ, the setS_x =φ⁻¹({x} ×${\displaystyle B_{⋔}}$) is called atransversal of the foliated chart. The set∂_τU =φ⁻¹(B_τ × (∂${\displaystyle B_{⋔}}$)) is called thetangential boundary ofU and${\displaystyle {\mathrm{\partial }}_{⋔}U}$ =φ⁻¹((∂B_τ) ×${\displaystyle B_{⋔}}$) is called thetransverse boundary ofU.^[7]

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

Afoliatedatlas of codimensionq and classC^r (0 ≤r ≤ ∞) on then-manifoldM is aC^r-atlas${\displaystyle \mathcal{U}=\{(U_{\alpha },{\phi }_{\alpha })\mid \alpha \in A\}}$ of foliated charts of codimensionq which arecoherently foliated in the sense that, wheneverP andQ are plaques in distinct charts of${\displaystyle \mathcal{U}}$, thenP ∩Q is open both inP andQ.^[8]

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

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

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

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

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

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

Onφ_β(U_α ∩U_β) the coordinates formula can be changed as^[9]

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

The condition that (U_α,x_α,y_α) and (U_β,x_β,y_β) be coherently foliated means that, ifP ⊂U_α is a plaque, the connected components ofP ∩U_β lie in (possibly distinct) plaques ofU_β. Equivalently, since the plaques ofU_α andU_β are level sets of the transverse coordinatesy_α andy_β, respectively, each pointz ∈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 ofx_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${\displaystyle \mathcal{V}}$ onM of the same codimension andsmoothness classC^r arecoherent${\displaystyle \left(\mathcal{U}\approx \mathcal{V}\right)}$ if${\displaystyle \mathcal{U}\cup \mathcal{V}}$ is a foliatedC^r-atlas. Coherence of foliated atlases is an equivalence relation.^[9]

Proof[9]

Reflexivity andsymmetry are immediate. To provetransitivity let${\displaystyle \mathcal{U}\approx \mathcal{V}}$ and${\displaystyle \mathcal{V}\approx \mathcal{W}}$. Let (Uα,xα,yα) ∈${\displaystyle \mathcal{U}}$ and (Wλ,xλ,yλ) ∈${\displaystyle \mathcal{W}}$ and suppose that there is a pointw ∈Uα ∩Wλ. Choose (Vδ,xδ,yδ) ∈${\displaystyle \mathcal{V}}$ such thatw ∈Vδ. By the above remarks, there is a neighborhoodN ofw inUα ∩Vδ ∩Wλ such that ${\displaystyle y_{\delta }=y_{\delta }(y_{\lambda })\text{on}{\phi }_{\lambda }(N),}$ ${\displaystyle y_{\alpha }=y_{\alpha }(y_{\delta })\text{on}{\phi }_{\delta }(N),}$ and hence ${\displaystyle y_{\alpha }=y_{\alpha }\left(y_{\delta }(y_{\lambda })\right)\text{on}{\phi }_{\delta }(N).}$ Sincew ∈Uα ∩Wλ is arbitrary, it can be concluded thatyα(xλ,yλ) is locally independent ofxλ. It is thus proven that${\displaystyle \mathcal{U}\approx \mathcal{W}}$, hence that coherence is transitive.[10]

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${\displaystyle \overline{U}}$ (theclosure ofU) is a subset ofW andφ =ψ|U then, if${\displaystyle \phi (U)=B_{\tau }\times B_{⋔},}$ it can be seen that${\displaystyle \psi |\overline{U}}$, written${\displaystyle \overline{\phi }}$, carries${\displaystyle \overline{U}}$ diffeomorphically onto${\displaystyle {\overline{B}}_{\tau }\times {\overline{B}}_{⋔}.}$

A foliated atlas is said to beregular if

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

By property (1), the coordinatesx_α andy_α extend to coordinates${\displaystyle {\overline{x}}_{\alpha }}$ and${\displaystyle {\overline{y}}_{\alpha }}$ on${\displaystyle {\overline{U}}_{\alpha }}$ and one writes${\displaystyle {\overline{\phi }}_{\alpha }=\left({\overline{x}}_{\alpha },{\overline{y}}_{\alpha }\right).}$ Property (3) is equivalent to requiring that, ifU_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

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

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 mapy_α can be viewed as asubmersion

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

and the formulasy_α =y_α(y_β) can be viewed asdiffeomorphisms

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

These satisfy thecocycle conditions. That is, ony_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 coherentrefinement that is regular.^[13]

Proof[13]

Fix a metric onM and a foliated atlas${\displaystyle \mathcal{W}.}$ Passing to asubcover, if necessary, one can assume that${\displaystyle \mathcal{W}={\left\{W_j,{\psi }_j\right\}}_{j=1}^l}$ is finite. Let ε > 0 be aLebesgue number for${\displaystyle \mathcal{W}.}$ That is, any subsetX ⊆M of diameter < ε lies entirely in someWj. For eachx ∈M, choosej such thatx ∈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 thatUx ⊂Wk,k ≠j, and writeψk = (xk,yk) as usual, whereyk :Wk →Rq is the transverse coordinate map. This is asubmersion having the plaques inWk as level sets. Thus,yk restricts to a submersionyk :Ux →Rq. This is locally constant inxj; so choosingUx smaller, if necessary, one can assume thatyk|${\displaystyle {\overline{U}}_x}$ has the plaques of${\displaystyle {\overline{U}}_x}$ as its level sets. That is, each plaque ofWk meets (hence contains) at most one (compact) plaque of${\displaystyle {\overline{U}}_x}$. Since 1 <k <l < ∞, one can chooseUx so that, wheneverUx ⊂Wk, distinct plaques of${\displaystyle {\overline{U}}_x}$ lie in distinct plaques ofWk. Pass to a finite subatlas${\displaystyle \mathcal{U}={\left\{U_i,{\phi }_i\right\}}_{i=1}^N}$ of {(Ux,φx) |x ∈M}. IfUi ∩Uj ≠ 0, then diam(Ui ∪Uj) < ε, and so there is an indexk 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 ofWk. 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. IfM is not compact, local compactness andsecond countability allows one to choose a sequence${\displaystyle {\left\{K_i\right\}}_{i=0}^{\mathrm{\infty }}}$ of compact subsets such thatKi ⊂ intKi+1 for eachi ≥ 0 and${\displaystyle M=\bigcup _{i=1}^{\mathrm{\infty }}{K}_i.}$ Passing to a subatlas, it is assumed that${\displaystyle \mathcal{W}={\left\{W_j,{\psi }_j\right\}}_{j=0}^{\mathrm{\infty }}}$ is countable and a strictly increasing sequence${\displaystyle {\left\{n_l\right\}}_{l=0}^{\mathrm{\infty }}}$ of positive integers can be found such that${\displaystyle {\mathcal{W}}_l={\left\{W_j,{\psi }_j\right\}}_{j=0}^{n_l}}$ coversKl. Let δl denote the distance fromKl to ∂Kl+1 and choose εl > 0 so small that εl < min{δl/2,εl-1} forl ≥ 1, ε0 < δ0/2, and εl is a Lebesgue number for${\displaystyle {\mathcal{W}}_l}$ (as an open cover ofKl) and for${\displaystyle {\mathcal{W}}_{l+1}}$ (as an open cover ofKl+1). More precisely, ifX ⊂M meetsKl (respectively,Kl+1) and diamX < εl, thenX lies in some element of${\displaystyle {\mathcal{W}}_l}$ (respectively,${\displaystyle {\mathcal{W}}_{l+1}}$). For eachx ∈Kl${\displaystyle ╲}$ intKl-1, construct (Ux,φx) as for the compact case, requiring that${\displaystyle {\overline{U}}_x}$ be a compact subset ofWj and thatφx =ψj|Ux, somej ≤nl. Also, require that diam${\displaystyle {\overline{U}}_x}$ < εl/2. As before, pass to a finite subcover${\displaystyle {\left\{U_i,{\phi }_i\right\}}_{i=n_{l-1}+1}^{n_l}}$ ofKl${\displaystyle ╲}$ intKl-1. (Here, it is takenn−1 = 0.) This creates a regular foliated atlas${\displaystyle \mathcal{U}={\left\{U_i,{\phi }_i\right\}}_{i=1}^{\mathrm{\infty }}}$ 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 throughdecomposition reaching to the following

Definition. Ap-dimensional, classC^r foliation of ann-dimensional manifoldM is a decomposition ofM into a union ofdisjoint connected submanifolds {L_α}_α∈A, called theleaves of the foliation, with the following property: Every point inM has a neighborhoodU and a system of local, classC^r coordinatesx=(x¹, ⋅⋅⋅,xⁿ) :U→Rⁿ such that for each leafL_α, the components ofU ∩L_α are described by the equationsx^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, ap-dimensional foliation${\displaystyle \mathcal{F}}$ of ann-manifoldM may be thought of as simply a collection{M_a} of pairwise-disjoint, connected, immersedp-dimensional submanifolds (the leaves of the foliation) ofM, such that for every pointx inM, there is a chart${\displaystyle (U,\phi )}$ withU homeomorphic toRⁿ containingx such that every leaf,M_a, meetsU in either the empty set or acountable collection of subspaces whose images under${\displaystyle \phi }$ in${\displaystyle \phi (M_a\cap U)}$ arep-dimensionalaffine subspaces whose firstn −p coordinates are constant.

Locally, every foliation is asubmersion allowing the following

Definition. LetM andQ be manifolds of dimensionn andq≤n respectively, and letf :M→Q be a submersion, that is, suppose that the rank of the function differential (theJacobian) isq. It follows from theImplicit Function Theorem thatƒ induces a codimension-q foliation onM where the leaves are defined to be the components off⁻¹(x) forx ∈Q.^[5]

This definition describes adimension-p foliation${\displaystyle \mathcal{F}}$ of ann-dimensional manifoldM that is a covered bychartsU_i together with maps

${\displaystyle {\phi }_i:U_i\to {\mathbb{R}}^n}$

such that for overlapping pairsU_i,U_j thetransition functionsφ_ij :Rⁿ →Rⁿ defined by

${\displaystyle {\phi }_{ij}={\phi }_j{\phi }_i^{-1}}$

take the form

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

wherex denotes the firstq =n −p coordinates, andy denotes the lastp co-ordinates. That is,

The splitting of the transition functionsφ_ij into${\displaystyle {\phi }_{ij}^1(x)}$ and${\displaystyle {\phi }_{ij}^2(x,y)}$ 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 codimensionq is associated to a unique foliation${\displaystyle \mathcal{F}}$ of codimensionq.^[13]

Proof[13]

Let${\displaystyle \mathcal{U}={\left\{U_a,{\phi }_{\alpha }\right\}}_{\alpha \in A}}$ be a regular foliated atlas of codimensionq. Define an equivalence relation onM by settingx ~y if and only if either there is a${\displaystyle \mathcal{U}}$-plaqueP0 such thatx,y ∈P0 or there is a sequenceL = {P0,P1,⋅⋅⋅,Pp} of${\displaystyle \mathcal{U}}$-plaques such thatx ∈P0, y ∈Pp, andPi ∩Pi-1 ≠ ∅ with 1 ≤i ≤p. The sequenceL will be called aplaque chain of length p connectingx andy. In the case thatx,y ∈P0, it is said that {P0} is a plaque chain of length 0 connectingx andy. The fact that ~ is an equivalence relation is clear. It is also clear that each equivalence classL 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 dimensionn −q. The open subsets of the plaquesP ⊂L form the base of a locally Euclidean topology onL of dimensionn −q andL is clearly connected in this topology. It is also trivial to check thatL isHausdorff. The main problem is to show thatL issecond countable. Since each plaque is 2nd countable, the same will hold forL if it is shown that the set of${\displaystyle \mathcal{U}}$-plaques inL is at most countably infinite. Fix one such plaqueP0. 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 lengthp of plaque chains that begin atP0, it is similarly proven that there are only finitely many of length ≤ p. Since every${\displaystyle \mathcal{U}}$-plaque inL is, by the definition of ~, reached by a finite plaque chain starting atP0, 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 Hausdorffp-dimensionalsubmanifolds. 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 onM 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 leafL is a union of${\displaystyle \mathcal{V}}$-plaques and of${\displaystyle \mathcal{U}}$-plaques. These plaques are open subsets in the manifold topology ofL, 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${\displaystyle \mathcal{V}\approx \mathcal{U}}$, letQ be a${\displaystyle \mathcal{V}}$-plaque. IfL is a leaf of${\displaystyle \mathcal{F}}$ andw ∈L ∩Q, letP ∈L be a${\displaystyle \mathcal{U}}$-plaque withw ∈P. ThenP ∩Q is an open neighborhood ofw inQ andP ∩Q ⊂L ∩Q. Sincew ∈L ∩Q is arbitrary, it follows thatL ∩Q is open inQ. SinceL is an arbitrary leaf, it follows thatQ decomposes into disjoint open subsets, each of which is the intersection ofQ with some leaf of${\displaystyle \mathcal{F}}$. SinceQ 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 onM and their associated foliated atlases induces a one-to-one correspondence between the set of foliations onM and the set of coherence classes of foliated atlases or, in other words, a foliation${\displaystyle \mathcal{F}}$ of codimensionq and classC^r onM is a coherence class of foliated atlases of codimensionq and classC^r onM.^[14] ByZorn's lemma, it is obvious that every coherence class of foliated atlases contains a unique maximal foliated atlas. Thus,

Definition. A foliation of codimensionq and classC^r onM is a maximal foliatedC^r-atlas of codimensionq onM.^[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 chartU_i, the stripesx =constant match up with the stripes on other chartsU_j. These submanifolds piece together from chart to chart to form maximalconnected injectivelyimmersed submanifolds called theleaves of the foliation.

If one shrinks the chartU_i it can be written asU_ix ×U_iy, whereU_ix ⊂R^n−p,U_iy ⊂R^p,U_iyis homeomorphic to the plaques, and the points ofU_ix parametrize the plaques inU_i. If one picksy₀ inU_iy, thenU_ix × {y₀} is a submanifold ofU_i that intersects every plaque exactly once. This is called a localtransversalsection of the foliation. Note that due tomonodromy global transversal sections of the foliation might not exist.

The caser = 0 is rather special. ThoseC⁰ foliations that arise in practice are usually "smooth-leaved". More precisely, they are of classC^r,0, in the following sense.

Definition. A foliation${\displaystyle \mathcal{F}}$ is of classC^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 classC^k, butx_α is of classC^r in the coordinatesx_β and its mixedx_β partials of orders ≤ r areC^k in the coordinates (x_β,y_β).^[14]

The above definition suggests the more general concept of afoliated space orabstract lamination. One relaxes the condition that the transversals be open, relatively compact subsets ofR^q, allowing the transverse coordinatesy_α to take their values in some more general topological spaceZ. The plaques are still open, relatively compact subsets ofR^p, the change of transverse coordinate formulay_α(y_β) is continuous andx_α(x_β,y_β) is of classC^r in the coordinatesx_β and its mixedx_β partials of orders ≤ r are continuous in the coordinates (x_β,y_β). One usually requiresM andZ 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. IfL is a leaf of${\displaystyle \mathcal{F}}$ ands is a path inL, one is interested in the behavior of the foliation in a neighborhood ofs inM. Intuitively, an inhabitant of the leaf walks along the paths, keeping an eye on all of the nearby leaves. As they (hereafter denoted bys(t)) proceed, some of these leaves may "peel away", getting out of visual range, others may suddenly come into range and approachL asymptotically, others may follow along in a more or less parallel fashion or wind aroundL laterally,etc. Ifs is a loop, thens(t) repeatedly returns to the same points(t₀) ast 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 theholonomy 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.

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

The term"first return (recurrence) map" comes from the theory of dynamical systems. Let Φ_t be a nonsingularC^r flow (r ≥ 1) on the compactn-manifoldM. In applications, one can imagine thatM is acyclotron or some closed loop with fluid flow. IfM 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 sectionN. That is,N is a compact, properly embedded,C^r submanifold ofM of dimensionn – 1, the foliation${\displaystyle \mathcal{F}}$ is transverse toN, and every flow line meetsN. Because the dimensions ofN and of the leaves are complementary, the transversality condition is that

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

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

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

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

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

Definef :N →N by the formulaf(y) = Φ_τ(y)(y). This is aC^r map. If the flow is reversed, exactly the same construction provides the inversef⁻¹; sof ∈ Diff^r(N). This diffeomorphism is the first return map and τ is called thefirst return time. While the first return time depends on the parametrization of the flow, it should be evident thatf depends only on the oriented foliation${\displaystyle \mathcal{F}}$. It is possible to reparametrize the flow Φ_t, keeping it nonsingular, of classC^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 thatM is the total space of a fiber bundle overS¹. Indeed, onR ×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 groupZ onR ×N defined by

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

for eachk ∈Z and for each (t,y) ∈R ×N. The mapping cylinder off is defined to be theC^r manifold

${\displaystyle M_f=(\mathbb{R}\times N)/{\sim }_f.}$

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

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

defined by the flow, induces a canonicalC^r diffeomorphism

${\displaystyle \phi :M_f\to M.}$

If we make the identificationM_f =M, then the projection ofR ×N ontoR induces aC^r map

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

that makesM into the total space of afiber bundle over the circle. This is just the projection ofS¹ ×D² ontoS¹. The foliation${\displaystyle \mathcal{F}}$ is transverse to the fibers of this bundle and the bundle projection π, restricted to each leafL, is a covering mapπ :L →S¹. This is called afoliated bundle.

Take as basepointx₀ ∈S¹ the equivalence class 0 +Z; so π⁻¹(x₀) is the original cross sectionN. For each loops onS¹, based atx₀, the homotopy class [s] ∈ π₁(S¹,x₀) is uniquely characterized by degs ∈Z. The loops lifts to a path in each flow line and it should be clear that the lifts_y that starts aty ∈N ends atf^k(y) ∈N, wherek = degs. The diffeomorphismf^k ∈ Diff^r(N) is also denoted byh_s and is called thetotal holonomy of the loops. Since this depends only on [s], this is a definition of a homomorphism

${\displaystyle h:{\pi }_1(S^1,x_0)\to {\mathrm{Diff}}^r(N),}$

called thetotal holonomy homomorphism for the foliated bundle.

Using fiber bundles in a more direct manner, let (M,${\displaystyle \mathcal{F}}$) be a foliatedn-manifold of codimensionq. Letπ :M →B be a fiber bundle withq-dimensional fiberF and connected base spaceB. Assume that all of these structures are of classC^r, 0 ≤r ≤ ∞, with the condition that, ifr = 0,B supports aC¹ structure. Since every maximalC¹ atlas onB contains aC^∞ subatlas, no generality is lost in assuming thatB is as smooth as desired. Finally, for eachx ∈B, assume that there is a connected, open neighborhoodU ⊆B ofx and a local trivialization

whereφ is aC^r diffeomorphism (a homeomorphism, ifr = 0) that carries$\mathcal{F}\mid {\pi }^{-1}(U)$ to the product foliation {U × {y}}_y ∈ F. Here,$\mathcal{F}\mid {\pi }^{-1}(U)$ is the foliation with leaves the connected components ofL ∩ π⁻¹(U), whereL ranges over the leaves of${\displaystyle \mathcal{F}}$. This is the general definition of the term "foliated bundle" (M,${\displaystyle \mathcal{F}}$,π) of classC^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 leafL of${\displaystyle \mathcal{F}}$ is a covering map π :L →B. In particular, each fiberF_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 ofB. A simple version of the problem is a foliation ofR², transverse to the fibration

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

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

but with infinitely many leaves missing they-axis. In the respective figure, it is intended that the "arrowed" leaves, and all above them, are asymptotic to the axisx = 0. One calls such a foliation incomplete relative to the fibration, meaning that some of the leaves "run off to infinity" as the parameterx ∈B approaches somex₀ ∈B. More precisely, there may be a leafL and a continuous paths : [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 ofL. This is analogous to the case of incomplete flows, where some flow lines "go to infinity" in finite time. Although such a leafL may elsewhere meet π⁻¹(x₀), it cannot evenly cover a neighborhood ofx₀, hence cannot be a covering space ofB underπ. WhenF is compact, however, it is true that transversality of${\displaystyle \mathcal{F}}$ to the fibration does guarantee completeness, hence that$(M,\mathcal{F},\pi )$ is a foliated bundle.

There is an atlas${\displaystyle \mathcal{U}}$ = {U_α,x_α}_α∈A onB, consisting of open, connected coordinate charts, together with trivializationsφ_α :π⁻¹(U_α) →U_α ×F that carry${\displaystyle \mathcal{F}}$|π⁻¹(U_α) to the product foliation. SetW_α =π⁻¹(U_α) and writeφ_α = (x_α,y_α) where (by abuse of notation)x_α representsx_α ∘π andy_α :π⁻¹(U_α) →F is the submersion obtained by composingφ_α with the canonical projectionU_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 ofW_α are the level sets ofy_α and this family of plaques is identical toF viay_α. SinceB is assumed to support aC^∞ structure, according to theWhitehead theorem one can fix a Riemannian metric onB 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 ofW_α meets exactly one plaque ofW_β. Then define aholonomy cocycle${\displaystyle \gamma ={\left\{{\gamma }_{\alpha \beta }\right\}}_{\alpha ,\beta \in A}}$ by setting

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

Consider ann-dimensional space, foliated as a product by subspaces consisting of points whose firstn −p coordinates are constant. This can be covered with a single chart. The statement is essentially thatRⁿ =R^n−p ×R^p with the leaves or plaquesR^p being enumerated byR^n−p. The analogy is seen directly in three dimensions, by takingn = 3 andp = 2: the 2-dimensional leaves of a book are enumerated by a (1-dimensional) page number.

A rather trivial example of foliations are productsM =B ×F, foliated by the leavesF_b = {b} ×F,b ∈B. (Another foliation ofM is given byB_f =B × { f } ,  f  ∈F.)

A more general class are flatG-bundles withG = Homeo(F) for a manifoldF. Given arepresentationρ :π₁(B) → Homeo(F), the flatHomeo(F)-bundle with monodromyρ is given by${\displaystyle M=\left(\tilde{B}\times F\right)/{\pi }_{1}B}$, whereπ₁(B) acts on theuniversal cover${\displaystyle \tilde{B}}$ bydeck transformations and onF by means of the representationρ.

Flat bundles fit into the framework offiber bundles. A mapπ :M →B between manifolds is a fiber bundle if there is a manifold F such that eachb ∈B has an open neighborhoodU such that there is a homeomorphism${\displaystyle \phi :{\pi }^{-1}(U)\to U\times F}$ with${\displaystyle \pi =p_1\phi }$, withp₁ :U ×F →U projection to the first factor. The fiber bundle yields a foliation by fibers${\displaystyle F_b:={\pi }^{-1}(\{b\}),b\in B}$. Its space of leaves L is homeomorphic toB, in particular L is a Hausdorff manifold.

IfM →N is acovering map between manifolds, andF is a foliation onN, then it pulls back to a foliation onM. More generally, if the map is merely abranched covering, where the branchlocus is transverse to the foliation, then the foliation can be pulled back.

IfMⁿ →N^q, (q ≤n) is asubmersion of manifolds, it follows from theinverse function theorem that the connected components of the fibers of the submersion define a codimensionq foliation ofM.Fiber bundles are an example of this type.

An example of a submersion, which is not a fiber bundle, is given by

${\displaystyle \{\begin{array}{l}f:[-1,1]\times \mathbb{R}\to \mathbb{R}\\ f(x,y)=(x^2-1)e^y\end{array}}$

This submersion yields a foliation of[−1, 1] ×R which is invariant under theZ-actions given by

${\displaystyle z(x,y)=(x,y+n),\text{or}z(x,y)=\left((-1{)}^{n}x,y\right)}$

for(x,y) ∈ [−1, 1] ×R andn ∈Z. The induced foliations ofZ \ ([−1, 1] ×R) are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliation (of the Möbius band). Their leaf spaces are not Hausdorff.

Define a submersion

${\displaystyle \{\begin{array}{l}f:D^n\times \mathbb{R}\to \mathbb{R}\\ f(r,\theta ,t):=(r^2-1)e^t\end{array}}$

where(r,θ) ∈ [0, 1] ×Sⁿ⁻¹ are cylindrical coordinates on then-dimensional diskDⁿ. This submersion yields a foliation ofDⁿ ×R which is invariant under theZ-actions given by

${\displaystyle z(x,y)=(x,y+z)}$

for(x,y) ∈Dⁿ ×R,z ∈Z. The induced foliation ofZ \ (Dⁿ ×R) is called then-dimensionalReeb foliation. Its leaf space is not Hausdorff.

Forn = 2, this gives a foliation of the solid torus which can be used to define theReeb foliation of the 3-sphere by gluing two solid tori along their boundary. Foliations of odd-dimensional spheresS²ⁿ⁺¹ are also explicitly known.^[16]

IfG is aLie group, andH is aLie subgroup, thenG is foliated bycosets ofH. WhenH isclosed inG, thequotient spaceG/H is a smooth (Hausdorff) manifold turningG into a fiber bundle with fiberH and baseG/H. This fiber bundle is actuallyprincipal, with structure groupH.

LetG be a Lie group acting smoothly on a manifoldM. If the action is alocally free action orfree action, then the orbits ofG define a foliation ofM.

If${\displaystyle \tilde{X}}$ is a nonsingular (i.e., nowhere zero) vector field, then the local flow defined by${\displaystyle \tilde{X}}$ patches together to define a foliation of dimension 1. Indeed, given an arbitrary pointx ∈M, the fact that${\displaystyle \tilde{X}}$ is nonsingular allows one to find a coordinate neighborhood (U,x¹,...,xⁿ) aboutx such that

and

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{x}^1}=\tilde{X}\mid U.}$

Geometrically, the flow lines of${\displaystyle \tilde{X}\mid U}$ are just the level sets

${\displaystyle x^i=c^i,2\le i\le n,}$

where all Since by convention manifolds are second countable, leaf anomalies like the "long line" are precluded by the second countability ofM itself. The difficulty can be sidestepped by requiring that${\displaystyle \tilde{X}}$ be a complete field (e.g., thatM be compact), hence that each leaf be a flow line.

An important class of 1-dimensional foliations on the torusT² are derived from projecting constant vector fields onT². A constant vector field

${\displaystyle \tilde{X}\equiv \left[\begin{array}{c}a\\ b\end{array}\right]}$

onR² is invariant by all translations inR², hence passes to a well-defined vector fieldX when projected on the torusT²=R²/Z². It is assumed thata ≠ 0. The foliation${\displaystyle \stackrel{\mathcal{~}}{\mathcal{F}}}$ onR² produced by${\displaystyle \tilde{X}}$ has as leaves the parallel straight lines of slope θ =b/a. This foliation is also invariant under translations and passes to the foliation${\displaystyle \mathcal{F}}$ onT² produced byX.

Each leaf of${\displaystyle \stackrel{\mathcal{~}}{\mathcal{F}}}$ is of the form

${\displaystyle \tilde{L}=\{(x_0+ta,y_0+tb){\}}_{t\in \mathbb{R}}.}$

If the slope isrational then all leaves are closed curveshomeomorphic to thecircle. In this case, one can takea,b ∈Z. For fixedt ∈R, the points of${\displaystyle \tilde{L}}$ corresponding to values oft ∈t₀ +Z all project to the same point ofT²; so the corresponding leafL of${\displaystyle \mathcal{F}}$ is an embedded circle inT². SinceL is arbitrary,${\displaystyle \mathcal{F}}$ is a foliation ofT² by circles. It follows rather easily that this foliation is actually a fiber bundle π :T² →S¹. This is known as alinear foliation.

When the slope θ =b/a isirrational, the leaves are noncompact, homeomorphic to the non-compactifiedreal line, anddense in the torus (cfIrrational rotation). The trajectory of each point (x₀,y₀) never returns to the same point, but generates an "everywhere dense" winding about the torus, i.e. approaches arbitrarily close to any given point. Thus the closure to the trajectory is the entire two-dimensional torus. This case is namedKronecker foliation, afterLeopold Kronecker and his

Kronecker's Density Theorem. If the real number θ is distinct from each rational multiple of π, then the set {e^inθ |n ∈Z} is dense in the unit circle.

Proof

To see this, note first that, if a leaf${\displaystyle \tilde{L}}$ of${\displaystyle \stackrel{\mathcal{~}}{\mathcal{F}}}$ does not project one-to-one intoT2, there must be a real numbert ≠ 0 such thatta andtb are both integers. But this would imply thatb/a ∈Q. In order to show that each leafL of${\displaystyle \mathcal{F}}$ is dense inT2, it is enough to show that, for everyv ∈R2, each leaf${\displaystyle \tilde{L}}$ of${\displaystyle \stackrel{\mathcal{~}}{\mathcal{F}}}$ achieves arbitrarily small positive distances from suitable points of the cosetv +Z2. A suitable translation inR2 allows one to assume thatv = 0; so the task is reduced to showing that${\displaystyle \tilde{L}}$ passes arbitrarily close to suitable points (n,m) ∈Z2. The line${\displaystyle \tilde{L}}$ has slope-intercept equation ${\displaystyle y=\theta x+c.}$ So it will suffice to find, for arbitrary η > 0, integersn andm such that Equivalently,c ∈R being arbitrary, one is reduced to showing that the set {θn −m}m,n∈Z is dense inR. This is essentially the criterion ofEudoxus that θ and 1 be incommensurable (i.e., that θ be irrational).

A similar construction using a foliation ofRⁿ by parallel lines yields a 1-dimensional foliation of then-torusRⁿ/Zⁿ associated with thelinear flow on the torus.

A flat bundle has not only its foliation by fibres but also a foliation transverse to the fibers, whose leaves are

${\displaystyle L_f:=\left\{p\left(\tilde{b},f\right):\tilde{b}\in \tilde{B}\right\},\text{ for }f\in F,}$

where${\displaystyle p:\tilde{B}\times F\to M}$ is the canonical projection. This foliation is called the suspension of the representationρ :π₁(B) → Homeo(F).

In particular, ifB =S¹ and${\displaystyle \phi :F\to F}$ is a homeomorphism ofF, then the suspension foliation of${\displaystyle \phi }$ is defined to be the suspension foliation of the representationρ :Z → Homeo(F) given byρ(z) = Φ^z. Its space of leaves isL =${\displaystyle \mathcal{F}}$/~, wherex ~y whenevery = Φⁿ(x) for somen ∈Z.

The simplest example of foliation by suspension is a manifoldX of dimensionq. Letf :X →X be a bijection. One defines the suspensionM =S¹ ×_fX as the quotient of [0,1] ×X by the equivalence relation (1,x) ~ (0,f(x)).

M =S¹ ×_fX = [0,1] ×X

Then automaticallyM carries two foliations:${\displaystyle \mathcal{F}}$₂ consisting of sets of the formF_2,t = {(t,x)_~ :x ∈X} and${\displaystyle \mathcal{F}}$₁ consisting of sets of the formF_2,x0 = {(t,x) :t ∈ [0,1] ,x ∈ O_x0}, where the orbit O_x0 is defined as

O_x0 = {...,f⁻²(x₀),f⁻¹(x₀),x₀,f(x₀),f²(x₀), ...},

where the exponent refers to the number of times the functionf is composed with itself. Note that O_x0 = O_f(x0) = O_f⁻²(x0), etc., so the same is true forF_1,x0. Understanding the foliation${\displaystyle \mathcal{F}}$₁ is equivalent to understanding the dynamics of the mapf. If the manifoldX is already foliated, one can use the construction to increase the codimension of the foliation, as long asf maps leaves to leaves.

The Kronecker foliations of the 2-torus are the suspension foliations of the rotationsR_α :S¹ →S¹ by angleα ∈ [0, 2π).

More specifically, if Σ = Σ₂ is the two-holed torus with C¹,C² ∈ Σ the two embedded circles let${\displaystyle \mathcal{F}}$ be the product foliation of the 3-manifoldM = Σ ×S¹ with leaves Σ × {y},y ∈S¹. Note thatN_i =C_i ×S¹ is an embedded torus and that${\displaystyle \mathcal{F}}$ is transverse toN_i,i = 1,2. Let Diff₊(S¹) denote the group of orientation-preserving diffeomorphisms ofS¹ and choosef₁,f₂ ∈ Diff₊(S¹). CutM apart alongN₁ andN₂, letting${\displaystyle N_i^{+}}$ and${\displaystyle N_i^{-}}$ denote the resulting copies ofN_i,i = 1,2. At this point one has a manifoldM' = Σ' ×S₁ with four boundary components${\displaystyle {\left\{N_i^{\pm }\right\}}_{i=1,2}.}$ The foliation${\displaystyle \mathcal{F}}$ has passed to a foliation${\displaystyle {\mathcal{F}}^{\mathrm{\prime }}}$ transverse to the boundary ∂M', each leaf of which is of the form Σ' × {y},y ∈S¹.

This leaf meets ∂M' in four circles${\displaystyle C_i^{\pm }\times \{y\}\subset N_i^{\pm }.}$ Ifz ∈C_i, the corresponding points in${\displaystyle C_i^{\pm }}$ are denoted byz^± and${\displaystyle N_i^{-}}$ is "reglued" to${\displaystyle N_i^{+}}$ by the identification

${\displaystyle (z^{-},y)\equiv (z^{+},f_i(y)),i=1,2.}$

Sincef₁ andf₂ are orientation-preserving diffeomorphisms ofS¹, they are isotopic to the identity and the manifold obtained by this regluing operation is homeomorphic toM. The leaves of${\displaystyle {\mathcal{F}}^{\mathrm{\prime }}}$, however, reassemble to produce a new foliation${\displaystyle \mathcal{F}}$(f₁,f₂) ofM. If a leafL of${\displaystyle \mathcal{F}}$(f₁,f₂) contains a piece Σ' × {y₀}, then

${\displaystyle L=\bigcup _{g\in G}{\mathrm{\Sigma }}^{\mathrm{\prime }}\times \{g(y_0)\},}$

whereG ⊂ Diff₊(S¹) is the subgroup generated by {f₁,f₂}. These copies of Σ' are attached to one another by identifications

(z⁻,g(y₀)) ≡ (z⁺,f₁(g(y₀))) for eachz ∈C₁,

(z⁻,g(y₀)) ≡ (z⁺,f₂(g(y₀))) for eachz ∈C₂,

whereg ranges overG. The leaf is completely determined by theG-orbit ofy₀ ∈S¹ and can he simple or immensely complicated. For instance, a leaf will be compact precisely if the correspondingG-orbit is finite. As an extreme example, ifG is trivial (f₁ =f₂ = id_S¹), then${\displaystyle \mathcal{F}}$(f₁,f₂) =${\displaystyle \mathcal{F}}$. If an orbit is dense inS¹, the corresponding leaf is dense inM. As an example, iff₁ andf₂ are rotations through rationally independent multiples of 2π, every leaf will be dense. In other examples, some leafL has closure${\displaystyle \overline{L}}$ that meets each factor {w} ×S¹ in aCantor set. Similar constructions can be made on Σ ×I, whereI is a compact, nondegenerate interval. Here one takesf₁,f₂ ∈ Diff₊(I) and, since ∂I is fixed pointwise by all orientation-preserving diffeomorphisms, one gets a foliation having the two components of ∂M as leaves. When one formsM' in this case, one gets a foliated manifold with corners. In either case, this construction is called thesuspension of a pair of diffeomorphisms and is a fertile source of interesting examples of codimension-one foliations.

There is a close relationship, assuming everything issmooth, withvector fields: given a vector fieldX onM that is never zero, itsintegral curves will give a 1-dimensional foliation. (i.e. a codimensionn − 1 foliation).

This observation generalises to theFrobenius theorem, saying that thenecessary and sufficient conditions for a distribution (i.e. ann −p dimensionalsubbundle of thetangent 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 underLie bracket. One can also phrase this differently, as a question ofreduction of the structure group of thetangent bundle fromGL(n) to a reducible subgroup.

The conditions in the Frobenius theorem appear asintegrability 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 asker(α), for some (non-canonical)α ∈ Ω¹ (i.e. a non-zero co-vector field). A givenα is integrable iffα ∧dα = 0 everywhere.

There is a global foliation theory, because topological constraints exist. For example, in thesurface case, an everywhere non-zero vector field can exist on anorientablecompact surface only for thetorus. This is a consequence of thePoincaré–Hopf index theorem, which shows theEuler characteristic will have to be 0. There are many deep connections withcontact topology, which is the "opposite" concept, requiring that the integrability condition isnever 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.

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