In mathematics, a p-dimensional foliation is a partition of a manifold into submanifolds, all of the same dimension p, locally modeled on the decomposition of Rn into the p-dimensional planes cut out by the equations . The submanifolds are called the leaves of the foliation.
The 3-sphere has a famous codimension-1 foliation called the Reeb foliation.
The submanifolds are required to be connected and injectively immersed, but they are not required to be embedded. For example, if m is a fixed irrational number, the torus is foliated by the set of straight lines in the torus of slope m. Each line is dense in the torus and is injectively immersed but not embedded.
If the manifold and the submanifolds are required to have a piecewise-linear, differentiable (of class Cr), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively.
The level sets of a smooth real-valued function on a manifold with no critical points form a codimension 1 foliation on the manifold. For example, in general relativity, spacetimes with some number of spatial dimensions and 1 time dimension are often foliated as the level sets of a smooth function whose gradient is timelike, so that the leaves are spacelike hypersurfaces. Every codimension 1 foliation locally arises this way, but generally does not arise this way globally. For example, 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.
Foliated charts and atlases In order to give a more precise definition of foliation, it is necessary to define some auxiliary elements.
A rectangular neighborhood in Rn is an open subset of the form B = J1 × ⋅⋅⋅ × Jn, where Ji 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
In the following definition, coordinate charts are considered that have values in Rp × Rq, 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 is a diffeomorphism, being a rectangular neighborhood in Rq and a rectangular neighborhood in Rp. The set Py = φ−1(Bτ × {y}), where , is called a plaque of this foliated chart. For each x ∈ Bτ, the set Sx = φ−1({x} × ) is called a transversal of the foliated chart. The set ∂τU = φ−1(Bτ × (∂)) is called the tangential boundary of U and = φ−1((∂Bτ) × ) is called the transverse boundary of U.
The foliated chart is the basic model for all foliations, the plaques being the leaves. The notation Bτ is read as "B-tangential" and as "B-transverse". There are also various possibilities. If both and Bτ 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 ∂ ≠ ∅ = ∂Bτ, then ∂U = ∂τU is a union of plaques and the foliation by plaques is tangent to the boundary. If ∂Bτ ≠ ∅ = ∂, then ∂U = is a union of transversals and the foliation is transverse to the boundary. Finally, if ∂ ≠ ∅ ≠ ∂Bτ, this is a model of a foliated manifold with a corner separating the tangential boundary from the transverse boundary.
A foliated atlas of codimension q and class Cr (0 ≤ r ≤ ∞) on the n-manifold M is a Cr-atlas of foliated charts of codimension q which are coherently foliated in the sense that, whenever P and Q are plaques in distinct charts of , then P ∩ Q is open both in P and Q.
A useful way to reformulate the notion of coherently foliated charts is to write for w ∈ Uα ∩ Uβ
The notation (Uα,φα) is often written (Uα,xα,yα), with
On φβ(Uα ∩ Uβ) the coordinates formula can be changed as
The condition that (Uα,xα,yα) and (Uβ,xβ,yβ) be coherently foliated means that, if P ⊂ Uα is a plaque, the connected components of P ∩ Uβ lie in (possibly distinct) plaques of Uβ. Equivalently, since the plaques of Uα and Uβ are level sets of the transverse coordinates yα and yβ, respectively, each point z ∈ Uα ∩ Uβ has a neighborhood in which the formula
is independent of xβ.
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 (Uα,φα) can meet multiple plaques of (Uβ,φβ). 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 and on M of the same codimension and smoothness class Cr are coherent if is a foliated Cr-atlas. Coherence of foliated atlases is an equivalence relation. {| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Proof
Reflexivity and symmetry are immediate. To prove transitivity let and . Let (Uα,xα,yα) ∈ and (Wλ,xλ,yλ) ∈ and suppose that there is a point w ∈ Uα ∩ Wλ. Choose (Vδ,xδ,yδ) ∈ such that w ∈ Vδ. By the above remarks, there is a neighborhood N of w in Uα ∩ Vδ ∩ Wλ such that
and hence
Since w ∈ Uα ∩ Wλ is arbitrary, it can be concluded that yα(xλ,yλ) is locally independent of xλ. It is thus proven that , hence that coherence is transitive.
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 (the closure of U) is a subset of W and φ = ψ|U then, if it can be seen that , written , carries diffeomorphically onto
A foliated atlas is said to be regular if for each α ∈ A, is a compact subset of a foliated chart (Wα,ψα) and φα = ψα|Uα; the cover {Uα | α ∈ A} is locally finite; if (Uα,φα) and (Uβ,φβ) are elements of the foliated atlas, then the interior of each closed plaque P ⊂ meets at most one plaque in
By property (1), the coordinates xα and yα extend to coordinates and on and one writes Property (3) is equivalent to requiring that, if Uα ∩ Uβ ≠ ∅, the transverse coordinate changes be independent of That is
has the formula
Similar assertions hold also for open charts (without the overlines). The transverse coordinate map yα can be viewed as a submersion
and the formulas yα = yα(yβ) can be viewed as diffeomorphisms
These satisfy the cocycle conditions. That is, on yδ(Uα ∩ Uβ ∩ Uδ),
and, in particular,
Using the above definitions for coherence and regularity it can be proven that every foliated atlas has a coherent refinement that is regular. {| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Proof
Fix a metric on M and a foliated atlas Passing to a subcover, if necessary, one can assume that is finite. Let ε > 0 be a Lebesgue number for 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 ⊆ ⊂ 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| has the plaques of as its level sets. That is, each plaque of Wk meets (hence contains) at most one (compact) plaque of . Since 1 < k < l < ∞, one can choose Ux so that, whenever Ux ⊂ Wk, distinct plaques of lie in distinct plaques of Wk. Pass to a finite subatlas of {(Ux,φx) | x ∈ M}. If Ui ∩ Uj ≠ 0, then diam(Ui ∪ Uj) < ε, and so there is an index k such that Distinct plaques of (respectively, of ) lie in distinct plaques of Wk. Hence each plaque of has interior meeting at most one plaque of and vice versa. By construction, is a coherent refinement of and is a regular foliated atlas.
If M is not compact, local compactness and second countability allows one to choose a sequence of compact subsets such that Ki ⊂ int Ki+1 for each i ≥ 0 and Passing to a subatlas, it is assumed that is countable and a strictly increasing sequence of positive integers can be found such that 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 (as an open cover of Kl) and for (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 (respectively, ). For each x ∈ Kl int Kl-1, construct (Ux,φx) as for the compact case, requiring that be a compact subset of Wj and that φx = ψj|Ux, some j ≤ nl. Also, require that diam < εl/2. As before, pass to a finite subcover of Kl int Kl-1. (Here, it is taken n−1 = 0.) This creates a regular foliated atlas that refines and is coherent with .
Foliation definitions 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 followingDefinition. A p-dimensional, class Cr foliation of an n-dimensional manifold M is a decomposition of M into a union of disjoint connected submanifolds {Lα}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 Cr coordinates x=(x1, ⋅⋅⋅, xn) : U→Rn such that for each leaf Lα, the components of U ∩ Lα are described by the equations xp+1=constant, ⋅⋅⋅, xn=constant. A foliation is denoted by ={Lα}α∈A.
The notion of leaves allows for an intuitive way of thinking about a foliation. For a slightly more geometrical definition, a -dimensional foliation of an -manifold may be thought of as simply a collection of pairwise-disjoint, connected, immersed -dimensional submanifolds (the leaves of the foliation) of , such that for every point in , there is a chart with homeomorphic to containing such that every leaf, , meets in either the empty set or a countable collection of subspaces whose images under in are -dimensional affine subspaces whose first coordinates are constant.
Locally, every foliation is a submersion allowing the followingDefinition. 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.
This definition describes a dimension- foliation of an -dimensional manifold that is a covered by charts together with maps
such that for overlapping pairs the transition functions defined by
take the form
where denotes the first = coordinates, and denotes the last co-ordinates. That is,
The splitting of the transition functions φij into and as a part of the submersion is completely analogous to the splitting of into and 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 of codimension q. {| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Proof