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In mathematics, a type of equivalence relation on an n manifold

In mathematics (differential geometry), a foliation is an equivalence relation on an n manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or 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. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1 (otherwise, C0 is a topological foliation). The number p (the dimension of the leaves) is called the dimension of the foliation and 1=q = n − p is called its codimension. In some papers on general relativity by mathematical physicists, the term foliation (or slicing) is used to describe a situation where the relevant Lorentz manifold (a (p+1) dimensional spacetime) has been decomposed into hypersurfaces of dimension p, specified as the level sets of a real valued smooth function (scalar field) whose gradient is everywhere non zero; this smooth function is moreover usually assumed to be a time function, meaning that its gradient is everywhere time 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 sometimes slices) of the foliation. 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 always locally the level sets of a function, they generally cannot be expressed this way globally,

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 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
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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

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"
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The notion of leaves allows for an intuitive way of thinking about a foliation. For a slightly more geometrical definition, p dimensional foliation of an n manifold M may be thought of as simply a collection {Ma} 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 with U homeomorphic to 'R'^(n) containing x such that every leaf, Ma, meets U in either the empty set or a countable collection of subspaces whose images under in 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. 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 Cr on M is a maximal foliated Cr atlas of codimension q on M.

Let (M, ) be a foliated manifold. If L is a leaf of 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.

Consider an n dimensional space, foliated as a product by subspaces consisting of points whose first n − p coordinates are constant. This can be covered with a single chart. The statement is essentially that with the leaves or plaques 'R'^(p) being enumerated by 'R'^(n−p). The analogy is seen directly in three dimensions, by taking and : the 2 dimensional leaves of a book are enumerated by a (1 dimensional) page number.

A rather trivial example of foliations are products , foliated by the leaves (Another foliation of M is given by .) A more general class are flat G bundles with for a manifold F. Given a representation , the flat Homeo(F) bundle with monodromy ρ is given by , where acts on the universal cover by deck transformations and on F by means of the representation ρ. Flat bundles fit into the framework of fiber bundles. A map between manifolds is a fiber bundle if there is a manifold F such that each b ∈ B has an open neighborhood U such that there is a homeomorphism with , with p1 : U × F → U projection to the first factor. The fiber bundle yields a foliation by fibers Its space of leaves L is homeomorphic to B, in particular L is a Hausdorff manifold.

If M → N is a covering map between manifolds, and F is a foliation on N, then it pulls back to a foliation on M. More generally, if the map is merely a branched covering, where the branch locus is transverse to the foliation, then the foliation can be pulled back.

If G is a Lie group, and H is a Lie subgroup, then G is foliated by cosets of H. When H is closed in G, the quotient space G/H is a smooth (Hausdorff) manifold turning G into a fiber bundle with fiber H and base G/H. This fiber bundle is actually principal, with structure group H.

Let G be a Lie group acting smoothly on a manifold M. If the action is a locally free action or free action, then the orbits of G define a foliation of M.

gave a necessary and sufficient condition for a distribution on a connected non compact manifold to be homotopic to an integrable distribution. showed that any compact manifold with a distribution has a foliation of the same dimension.