{"history":[],"article":{"id":"3146218d-9820-421f-8718-3b45f2d37e59","title":"Foliation","slug":"foliation","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics) ([differential geometry](https://bliptext.com/articles/differential-geometry)), a **foliation** is an [equivalence relation](https://bliptext.com/articles/equivalence-relation) on an n-manifold, the [equivalence classes](https://bliptext.com/articles/equivalence-class) being connected, [injectively](https://bliptext.com/articles/injective-function) [immersed submanifolds](https://bliptext.com/articles/immersed-submanifold), all of the same dimension p, modeled on the [decomposition](https://bliptext.com/articles/manifold-decomposition) of the [real coordinate space](https://bliptext.com/articles/real-coordinate-space) ******R******n into the [cosets](https://bliptext.com/articles/coset) x + ******R******p of the standardly [embedded](https://bliptext.com/articles/embedding) [subspace](https://bliptext.com/articles/subspace-topology) ******R******p. The [equivalence classes](https://bliptext.com/articles/equivalence-class) are called the **leaves** of the **foliation**.\nIf the manifold and/or the submanifolds are required to have a [piecewise-linear](https://bliptext.com/articles/piecewise-linear-manifold), [differentiable](https://bliptext.com/articles/differentiable-manifold) (of class Cr), or [analytic](https://bliptext.com/articles/analytic-manifold) structure then one defines [piecewise-linear](https://bliptext.com/articles/piecewise-linear-manifold), [differentiable](https://bliptext.com/articles/differentiable-manifold), or [analytic](https://bliptext.com/articles/analytic-manifold) foliations, respectively.\nIn the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1 (otherwise, C0 is a topological foliation).\nThe number p (the dimension of the leaves) is called the dimension of the foliation and q = n − p is called its [codimension](https://bliptext.com/articles/codimension).\nIn some papers on [general relativity](https://bliptext.com/articles/general-relativity) by mathematical physicists, the term foliation (or **slicing**) is used to describe a situation where the relevant [Lorentz manifold](https://bliptext.com/articles/pseudo-riemannian-manifold) (a (p+1)-dimensional [spacetime](https://bliptext.com/articles/spacetime)) has been decomposed into [hypersurfaces](https://bliptext.com/articles/hypersurface) of dimension p, specified as the level sets of a real-valued [smooth function](https://bliptext.com/articles/smooth-function) ([scalar field](https://bliptext.com/articles/scalar-field)) whose [gradient](https://bliptext.com/articles/gradient) is everywhere non-zero; this [smooth function](https://bliptext.com/articles/smooth-function) is moreover usually assumed to be a **time function,** meaning that its [gradient](https://bliptext.com/articles/gradient) is everywhere [time-like](https://bliptext.com/articles/spacetime), so that its level-sets are all space-like [hypersurfaces](https://bliptext.com/articles/hypersurface).\nIn deference to standard mathematical terminology, these hypersurface are often called the leaves (or sometimes **slices**) of the foliation.\nNote 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, 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.\nFor example, while the [3-sphere](https://bliptext.com/articles/3-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.\n# [2-dimensional section of Reeb foliation | upload.wikimedia.org/wikipedia/commons/c/cb/Reebfoliation-ring-2d-2.svg]\n# [3-dimensional model of Reeb foliation | upload.wikimedia.org/wikipedia/commons/c/cc/Reeb///foliation///half-torus///POV-Ray.png]\n\n# Foliated charts and atlases\nIn order to give a more precise definition of foliation, it is necessary to define some auxiliary elements.\nA rectangular [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) in **R**n is an [open](https://bliptext.com/articles/open-set) [subset](https://bliptext.com/articles/subset) of the form B = J1 × ⋅⋅⋅ × Jn, where Ji is a (possibly unbounded) relatively [open](https://bliptext.com/articles/open-set) interval in the ith coordinate axis.\nIf J1 is of the form (a,0], it is said that B has [boundary](https://bliptext.com/articles/boundary-topology) In the following definition, coordinate charts are considered that have values in ****R****p × ****R****q, allowing the possibility of manifolds with [boundary](https://bliptext.com/articles/boundary-topology) and ([convex](https://bliptext.com/articles/convex-set)) corners.\nA foliated chart on the n-manifold M of codimension q is a pair (U,φ), where U ⊆ M is open and is a [diffeomorphism](https://bliptext.com/articles/diffeomorphism), being a rectangular neighborhood in ****R****q and B_{\\tau} a rectangular neighborhood in ****R****p. The set Py = φ−1(Bτ × {y}), where, is called a plaque of this foliated chart.\nFor each x ∈ Bτ, the set Sx = φ−1({x} × ) is called a [transversal](https://bliptext.com/articles/transversal-geometry) of the foliated chart.\nThe set ∂τU = φ−1(Bτ × (∂)) is called the tangential boundary of U and = φ−1((∂Bτ) × ) is called the transverse boundary of U.\nThe foliated chart is the basic model for all foliations, the plaques being the leaves.\nThe notation Bτ is read as \"B-tangential\" and as \"B-transverse\".\nThere are also various possibilities.\nIf both and Bτ have empty boundary, the foliated chart models codimension-q foliations of n-manifolds without boundary.\nIf 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.\nSpecifically, if ∂ ≠ ∅ = ∂Bτ, then ∂U = ∂τU is a union of plaques and the foliation by plaques is tangent to the boundary.\nIf ∂Bτ ≠ ∅ = ∂, then ∂U = is a union of transversals and the foliation is transverse to the boundary.\nFinally, if ∂ ≠ ∅ ≠ ∂Bτ, this is a model of a foliated manifold with a corner separating the tangential boundary from the transverse boundary.\nA foliated [atlas](https://bliptext.com/articles/atlas-topology) of codimension q and class Cr (0 ≤ r ≤ ∞) on the n-manifold M is a Cr-[atlas](https://bliptext.com/articles/atlas-topology) of foliated charts of codimension q which are coherently foliated in the sense that, whenever P and Q are plaques in distinct charts of \\mathcal{U}, then P ∩ Q is open both in P and Q.\nA useful way to reformulate the notion of coherently foliated charts is to write for w ∈ Uα ∩ Uβ\nThe notation (Uα,φα) is often written (Uα,xα,yα), with\nOn 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.\nEquivalently, 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\nis independent of xβ.\nThe main use of foliated atlases is to link their overlapping plaques to form the leaves of a foliation.\nFor this and other purposes, the general definition of foliated atlas above is a bit clumsy.\nOne problem is that a plaque of (Uα,φα) can meet multiple plaques of (Uβ,φβ).\nIt can even happen that a plaque of one chart meets infinitely many plaques of another chart.\nHowever, no generality is lost in assuming the situation to be much more regular as shown below.\nTwo foliated atlases \\mathcal{U} and \\mathcal{V} on M of the same codimension and [smoothness](https://bliptext.com/articles/smoothness) class Cr are coherent if is a foliated Cr-atlas.\nCoherence of foliated atlases is an equivalence relation.\n!Proof\n[Reflexivity](https://bliptext.com/articles/reflexive-relation) and [symmetry](https://bliptext.com/articles/symmetric-relation) are immediate.\nTo prove [transitivity](https://bliptext.com/articles/transitive-relation) let and.\nLet (Uα,xα,yα) ∈ \\mathcal{U} and (Wλ,xλ,yλ) ∈ \\mathcal{W} and suppose that there is a point w ∈ Uα ∩ Wλ.\nChoose (Vδ,xδ,yδ) ∈ \\mathcal{V} such that w ∈ Vδ.\nBy the above remarks, there is a neighborhood N of w in Uα ∩ Vδ ∩ Wλ such that\nand hence\nSince w ∈ Uα ∩ Wλ is arbitrary, it can be concluded that yα(xλ,yλ) is locally independent of xλ.\nIt is thus proven that, hence that coherence is transitive.\nPlaques and transversals defined above on open sets are also open.\nBut one can speak also of closed plaques and transversals.\nNamely, if (U,φ) and (W,ψ) are foliated charts such that (the [closure](https://bliptext.com/articles/closure-mathematics) of U) is a subset of W and φ = ψ|U then, if it can be seen that, written , carries diffeomorphically onto\nA foliated atlas is said to be regular if\nBy 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\nSimilar assertions hold also for open charts (without the overlines).\nThe transverse coordinate map yα can be viewed as a [submersion](https://bliptext.com/articles/submersion-mathematics)\nand the formulas yα = yα(yβ) can be viewed as [diffeomorphisms](https://bliptext.com/articles/diffeomorphisms)\nThese satisfy the [cocycle conditions](https://bliptext.com/articles/cocycle-condition).\nThat is, on yδ(Uα ∩ Uβ ∩ Uδ),\nand, in particular,\nUsing the above definitions for coherence and regularity it can be proven that every foliated atlas has a coherent [refinement](https://bliptext.com/articles/refinement-topology) that is regular.\n!Proof\nFix a metric on M and a foliated atlas Passing to a [subcover](https://bliptext.com/articles/cover-topology), if necessary, one can assume that is finite.\nLet ε > 0 be a [Lebesgue number](https://bliptext.com/articles/lebesgue-number-lemma) for That is, any subset X ⊆ M of diameter < ε lies entirely in some Wj.\nFor each x ∈ M, choose j such that x ∈ Wj and choose a foliated chart (Ux, φx) such that\nSuppose that Ux ⊂ Wk, k ≠ j, and write ψk = (xk,yk) as usual, where yk : Wk → **R**q is the transverse coordinate map.\nThis is a [submersion](https://bliptext.com/articles/submersion-mathematics) having the plaques in Wk as level sets.\nThus, yk restricts to a submersion yk : Ux → **R**q.\nThis is locally constant in xj; so choosing Ux smaller, if necessary, one can assume that yk| has the plaques of as its level sets.\nThat is, each plaque of Wk meets (hence contains) at most one (compact) plaque of.\nSince 1 < k < l < ∞, one can choose Ux so that, whenever Ux ⊂ Wk, distinct plaques of lie in distinct plaques of Wk.\nPass to a finite subatlas of {(Ux,φx) | x ∈ M}.\nIf 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.\nHence each plaque of has interior meeting at most one plaque of and vice versa.\nBy construction, \\mathcal{U} is a coherent refinement of \\mathcal{W} and is a regular foliated atlas.\nIf M is not compact, local compactness and [second countability](https://bliptext.com/articles/second-countable-space) 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.\nLet 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).\nMore precisely, if X ⊂ M meets Kl (respectively, Kl+1) and diam X < εl, then X lies in some element of (respectively, ).\nFor each x ∈ Kl \\diagdown 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.\nAlso, require that diam < εl/2.\nAs before, pass to a finite subcover of Kl \\diagdown int Kl-1.\n(Here, it is taken n−1 = 0.) This creates a regular foliated atlas that refines \\mathcal{W} and is coherent with.\n# [200px | upload.wikimedia.org/wikipedia/commons/8/80/Foliated///chart.svg]\n# [(a) Foliation tangent to the boundary ∂ ≠ ∅ = ∂Bτ; (b) Foliation transverse to the boundary ∂Bτ ≠ ∅ = ∂; (c) Foliation with a corner separating the tangential boundary from the transverse boundary ∂ ≠ ∅ ≠ ∂Bτ. | upload.wikimedia.org/wikipedia/commons/3/32/Foliated///chart///boundaries.svg]\n# [Plaques of Uα each meet two plaques of Uβ. | upload.wikimedia.org/wikipedia/commons/e/e3/Two///plaques///meeting.svg]\n# [Sample charts in a regular foliated atlas. | upload.wikimedia.org/wikipedia/commons/d/dc/Regular///foliated///atlas.svg]\n\n# Foliation definitions\nSeveral alternative definitions of foliation exist depending on the way through which the foliation is achieved.\nThe most common way to achieve a foliation is through [decomposition](https://bliptext.com/articles/manifold-decomposition) reaching to the following\n**Definition.** A p-dimensional, class Cr foliation of an n-dimensional manifold M is a decomposition of M into a union of [disjoint](https://bliptext.com/articles/disjoint-sets) 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→**R**n such that for each leaf Lα, the components of U ∩ Lα are described by the equations xp+1=constant, ⋅⋅⋅, xn=constant.\nA foliation is denoted by \\mathcal{F}={Lα}α∈A.\nThe notion of leaves allows for an intuitive way of thinking about a foliation.\nFor a slightly more geometrical definition, p-dimensional foliation \\mathcal{F} of an n-manifold M may be thought of as simply a collection\n{Ma}\nof 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 (U,\\varphi) with U homeomorphic to\nRn\ncontaining x such that every leaf,\nMa\n, meets U in either the empty set or a [countable](https://bliptext.com/articles/countable) collection of subspaces whose images under \\varphi in are p-dimensional [affine subspaces](https://bliptext.com/articles/affine-space) whose first\nn − p\ncoordinates are constant.\nLocally, every foliation is a [submersion](https://bliptext.com/articles/submersion-mathematics) allowing the following\n**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](https://bliptext.com/articles/jacobian-matrix-and-determinant)) is q.\nIt follows from the [Implicit Function Theorem](https://bliptext.com/articles/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.\nThis definition describes a [dimension](https://bliptext.com/articles/dimension)-p foliation \\mathcal{F} of an n-[dimension](https://bliptext.com/articles/dimension)al manifold M that is a covered by [charts](https://bliptext.com/articles/chart-topology)\nUi\ntogether with maps\nsuch that for overlapping pairs\nUi, Uj\nthe [transition functions](https://bliptext.com/articles/transition-map)\nφij : Rn → Rn\ndefined by\ntake the form\nwhere x denotes the first q =\nn − p\ncoordinates, and y denotes the last p co-ordinates.\nThat is,\nThe 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.\nThis makes possible another definition of foliations in terms of regular foliated atlases.\nTo this end, one has to prove first that every regular foliated atlas of codimension q is associated\nto a unique foliation \\mathcal{F} of codimension q.\n!Proof\nLet be a regular foliated atlas of codimension q.\nDefine an equivalence relation on M by setting x ~ y if and only if either there is a \\mathcal{U}-plaque P0 such that x,y ∈ P0 or there is a sequence L = {P0,P1,⋅⋅⋅,Pp} of \\mathcal{U}-plaques such that x ∈ P0, y ∈ Pp, and Pi ∩ Pi-1 ≠ ∅ with 1 ≤ i ≤ p.\nThe sequence L will be called a plaque chain of length p connecting x and y.\nIn the case that x,y ∈ P0, it is said that {P0} is a plaque chain of length 0 connecting x and y.\nThe fact that ~ is an equivalence relation is clear.\nIt is also clear that each equivalence class L is a union of plaques.\nSince \\mathcal{U}-plaques can only overlap in open subsets of each other, L is locally a topologically immersed submanifold of dimension n − q.\nThe 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.\nIt is also trivial to check that L is [Hausdorff](https://bliptext.com/articles/hausdorff-space).\nThe main problem is to show that L is [second countable](https://bliptext.com/articles/second-countable-space).\nSince each plaque is 2nd countable, the same will hold for L if it is shown that the set of \\mathcal{U}-plaques in L is at most countably infinite.\nFix one such plaque P0.\nBy the definition of a regular, foliated atlas, P0 meets only finitely many other plaques.\nThat is, there are only finitely many plaque chains {P0,Pi} of length 1.\nBy 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 \\mathcal{U}-plaque in L is, by the definition of ~, reached by a finite plaque chain starting at P0, the assertion follows.\nAs 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](https://bliptext.com/articles/submanifold).\nNext, 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.\nMore specifically, if \\mathcal{U} and \\mathcal{V} are foliated atlases on M and if \\mathcal{U} is associated to a foliation \\mathcal{F} then \\mathcal{U} and \\mathcal{V} are coherent if and only if \\mathcal{V} is also associated to \\mathcal{F}.\n!Proof\nIf \\mathcal{V} is also associated to \\mathcal{F}, every leaf L is a union of \\mathcal{V}-plaques and of \\mathcal{U}-plaques.\nThese plaques are open subsets in the manifold topology of L, hence intersect in open subsets of each other.\nSince plaques are connected, a \\mathcal{U}-plaque cannot intersect a \\mathcal{V}-plaque unless they lie in a common leaf; so the foliated atlases are coherent.\nConversely, if we only know that \\mathcal{U} is associated to \\mathcal{F} and that, let Q be a \\mathcal{V}-plaque.\nIf L is a leaf of \\mathcal{F} and w ∈ L ∩ Q, let P ∈ L be a \\mathcal{U}-plaque with w ∈ P.\nThen P ∩ Q is an open neighborhood of w in Q and P ∩ Q ⊂ L ∩ Q.\nSince w ∈ L ∩ Q is arbitrary, it follows that L ∩ Q is open in Q.\nSince 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 \\mathcal{F}.\nSince Q is connected, L ∩ Q = Q.\nFinally, Q is an arbitrary \\mathcal{V}-plaque, and so \\mathcal{V} is associated to \\mathcal{F}.\nIt 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 \\mathcal{F} of codimension q and class Cr on M is a coherence class of foliated atlases of codimension q and class Cr on M.\nBy [Zorn's lemma](https://bliptext.com/articles/zorn-s-lemma), it is obvious that every coherence class of foliated atlases contains a unique maximal foliated atlas.\nThus,\n**Definition.** A foliation of codimension q and class Cr on M is a maximal foliated Cr-atlas of codimension q on M.\nIn practice, a relatively small foliated atlas is generally used to represent a foliation.\nUsually, it is also required this atlas to be regular.\nIn the chart\nUi\n, the stripes\nmatch up with the stripes on other charts\nUj\n. These submanifolds piece together from chart to chart to form maximal [connected](https://bliptext.com/articles/connected-space) injectively [immersed submanifolds](https://bliptext.com/articles/immersed-submanifold) called the **leaves** of the foliation.\nIf one shrinks the chart\nUi\nit can be written as\nUix × Uiy\n, where\nUix ⊂ Rn−p, Uiy ⊂ Rp\n,\nUiy\nis homeomorphic to the plaques, and the points of\nUix\nparametrize the plaques in\nUi\n. If one picks\ny0\nin\nUiy\n, then\nUix × {y0}\nis a submanifold of\nUi\nthat intersects every plaque exactly once.\nThis is called a local transversal section of the foliation.\nNote that due to [monodromy](https://bliptext.com/articles/monodromy) global transversal sections of the foliation might not exist.\nThe case r = 0 is rather special.\nThose C0 foliations that arise in practice are usually \"smooth-leaved\".\nMore precisely, they are of class Cr,0, in the following sense.\n**Definition.** A foliation \\mathcal{F} is of class Cr,k, r > k ≥ 0, if the corresponding coherence class of foliated atlases contains a regular foliated atlas {Uα,xα,yα}α∈A such that the change of coordinate formula\nis of class Ck, but xα is of class Cr in the coordinates xβ and its mixed xβ partials of orders ≤ r are Ck in the coordinates (xβ,yβ).\nThe above definition suggests the more general concept of a foliated space or abstract lamination.\nOne relaxes the condition that the transversals be open, relatively compact subsets of **R**q, allowing the transverse coordinates yα to take their values in some more general topological space Z.\nThe plaques are still open, relatively compact subsets of **R**p, the change of transverse coordinate formula yα(yβ) is continuous and xα(xβ,yβ) is of class Cr in the coordinates xβ and its mixed xβ partials of orders ≤ r are continuous in the coordinates (xβ,yβ).\nOne usually requires M and Z to be locally compact, [second countable](https://bliptext.com/articles/second-countable) and metrizable.\nThis may seem like a rather wild generalization, but there are contexts in which it is useful.\n# [Decomposition through the coordinates function x : U→Rn. | upload.wikimedia.org/wikipedia/commons/f/fe/Foliation///definition.svg]\n\n# Holonomy\nLet (M, \\mathcal{F}) be a foliated manifold.\nIf L is a leaf of \\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.\nIntuitively, an inhabitant of the leaf walks along the path s, keeping an eye on all of the nearby leaves.\nAs 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.\nIf 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.\nThis behavior, when appropriately formalized, is called the holonomy of the foliation.\nHolonomy 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.\n\n## Foliated bundles\nThe easiest case of holonomy to understand is the total holonomy of a foliated bundle.\nThis is a generalization of the notion of a [Poincaré map](https://bliptext.com/articles/poincar-map).\nThe term [\"first return (recurrence) map\"](https://bliptext.com/articles/poincar-map) comes from the theory of dynamical systems.\nLet Φt be a nonsingular Cr flow (r ≥ 1) on the compact n-manifold M.\nIn applications, one can imagine that M is a [cyclotron](https://bliptext.com/articles/cyclotron) or some closed loop with fluid flow.\nIf M has a boundary, the flow is assumed to be tangent to the boundary.\nThe flow generates a 1-dimensional foliation \\mathcal{F}.\nIf one remembers the positive direction of flow, but otherwise forgets the parametrization (shape of trajectory, velocity, etc.), the underlying foliation \\mathcal{F} is said to be oriented.\nSuppose that the flow admits a global cross section N.\nThat is, N is a compact, properly embedded, Cr submanifold of M of dimension n – 1, the foliation \\mathcal{F} is transverse to N, and every flow line meets N.\nBecause the dimensions of N and of the leaves are complementary, the transversality condition is that\nLet y ∈ N and consider the ω-[limit set](https://bliptext.com/articles/limit-set) ω(y) of all accumulation points in M of all sequences, where tk goes to infinity.\nIt can be shown that ω(y) is compact, nonempty, and a union of flow lines.\nIf there is a value t* ∈ **R** such that Φt*(z) ∈ N and it follows that\nSince N is compact and \\mathcal{F} is transverse to N, it follows that the set {t > 0 | Φt(y) ∈ N} is a monotonically increasing sequence that diverges to infinity.\nAs y ∈ N varies, let τ(y) = τ1(y), defining in this way a positive function τ ∈ Cr(N) (the first return time) such that, for arbitrary y ∈ N, Φt(y) ∉ N, 0 < t < τ(y), and Φτ(y)(y) ∈ N.\nDefine f : N → N by the formula f(y) = Φτ(y)(y).\nThis is a Cr map.\nIf the flow is reversed, exactly the same construction provides the inverse f−1; so f ∈ Diffr(N).\nThis diffeomorphism is the first return map and τ is called the [first return time](https://bliptext.com/articles/first-return-time).\nWhile the first return time depends on the parametrization of the flow, it should be evident that f depends only on the oriented foliation \\mathcal{F}.\nIt is possible to reparametrize the flow Φt, keeping it nonsingular, of class Cr, and not reversing its direction, so that τ ≡ 1.\nThe 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.\nIndeed, on ******R****** × N, define ~f to be the equivalence relation generated by Equivalently, this is the orbit equivalence for the action of the additive group ****Z**** on ******R****** × N defined by for each k ∈ ****Z**** and for each (t,y) ∈ ******R****** × N.\nThe mapping cylinder of f is defined to be the Cr manifold\nBy 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\ndefined by the flow, induces a canonical Cr diffeomorphism\nIf we make the identification Mf = M, then the projection of ****R**** × N onto ****R**** induces a Cr map\nthat makes M into the total space of a [fiber bundle](https://bliptext.com/articles/fiber-bundle) over the circle.\nThis is just the projection of S1 × D2 onto S1.\nThe foliation \\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.\nThis is called a foliated bundle.\nTake as basepoint x0 ∈ S1 the equivalence class 0 + **Z**; so π−1(x0) is the original cross section N.\nFor each loop s on S1, based at x0, the homotopy class [s] ∈ π1(S1,x0) is uniquely characterized by deg s ∈ **Z**.\nThe loop s lifts to a path in each flow line and it should be clear that the lift sy that starts at y ∈ N ends at fk(y) ∈ N, where k = deg s.\nThe diffeomorphism fk ∈ Diffr(N) is also denoted by hs and is called the total holonomy of the loop s.\nSince this depends only on [s], this is a definition of a homomorphism\ncalled the total holonomy homomorphism for the foliated bundle.\nUsing fiber bundles in a more direct manner, let (M,\\mathcal{F}) be a foliated n-manifold of codimension q.\nLet π : M → B be a fiber bundle with q-dimensional fiber F and connected base space B.\nAssume that all of these structures are of class Cr, 0 ≤ r ≤ ∞, with the condition that, if r = 0, B supports a C1 structure.\nSince every maximal C1 atlas on B contains a C∞ subatlas, no generality is lost in assuming that B is as smooth as desired.\nFinally, for each x ∈ B, assume that there is a connected, open neighborhood U ⊆ B of x and a local trivialization\nwhere φ is a Cr diffeomorphism (a homeomorphism, if r = 0) that carries to the product foliation {U × {y}}y ∈ F. Here, is the foliation with leaves the connected components of L ∩ π−1(U), where L ranges over the leaves of \\mathcal{F}.\nThis is the general definition of the term \"foliated bundle\" (M,\\mathcal{F},π) of class Cr.\n\\mathcal{F} is transverse to the fibers of π (it is said that \\mathcal{F} is transverse to the fibration) and that the restriction of π to each leaf L of \\mathcal{F} is a covering map π : L → B.\nIn particular, each fiber Fx = π−1(x) meets every leaf of \\mathcal{F}.\nThe fiber is a cross section of \\mathcal{F} in complete analogy with the notion of a cross section of a flow.\nThe foliation \\mathcal{F} being transverse to the fibers does not, of itself, guarantee that the leaves are covering spaces of B.\nA simple version of the problem is a foliation of **R**2, transverse to the fibration\nbut with infinitely many leaves missing the y-axis.\nIn the respective figure, it is intended that the \"arrowed\" leaves, and all above them, are asymptotic to the axis x = 0.\nOne 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.\nMore precisely, there may be a leaf L and a continuous path s : [0,a) → L such that limt→a−π(s(t)) = x0 ∈ B, but limt→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 \\mathcal{F} to the fibration does guarantee completeness, hence that is a foliated bundle.\nThere is an atlas \\mathcal{U} = {Uα,xα}α∈A on B, consisting of open, connected coordinate charts, together with trivializations φα : π−1(Uα) → Uα × F that carry \\mathcal{F}|π−1(Uα) to the product foliation.\nSet Wα = π−1(Uα) and write φα = (xα,yα) where (by abuse of notation) xα represents xα ∘ π and yα : π−1(Uα) → F is the submersion obtained by composing φα with the canonical projection Uα × F → F.\nThe atlas \\mathcal{W} = {Wα,xα,yα}α∈A plays a role analogous to that of a foliated atlas.\nThe plaques of Wα are the level sets of yα and this family of plaques is identical to F via yα.\nSince B is assumed to support a C∞ structure, according to the [Whitehead theorem](https://bliptext.com/articles/whitehead-theorem) one can fix a Riemannian metric on B and choose the atlas \\mathcal{U} to be geodesically convex.\nThus, Uα ∩ Uβ is always connected.\nIf this intersection is nonempty, each plaque of Wα meets exactly one plaque of Wβ.\nThen define a holonomy cocycle by setting\n# [300px | upload.wikimedia.org/wikipedia/commons/2/27/Holonomy.svg]\n\n# Examples\n\n## Flat space\nConsider an n-dimensional space, foliated as a product by subspaces consisting of points whose first\nn − p\ncoordinates are constant.\nThis can be covered with a single chart.\nThe statement is essentially that\nwith the leaves or plaques\nRp\nbeing enumerated by\nRn−p\n. The analogy is seen directly in three dimensions, by taking\nand\n\n## Bundles\nA rather trivial example of foliations are products\n, foliated by the leaves\n. (Another foliation of M is given by\n.)\nA more general class are flat G-bundles with\nfor a manifold F. Given a [representation](https://bliptext.com/articles/representation-mathematics)\nρ : π1(B) → Homeo(F)\n, the flat\nHomeo(F)\n-bundle with monodromy ρ is given by, where\nπ1(B)\nacts on the [universal cover](https://bliptext.com/articles/universal-cover) by [deck transformations](https://bliptext.com/articles/covering-space) and on F by means of the representation ρ.\nFlat bundles fit into the framework of [fiber bundles](https://bliptext.com/articles/fiber-bundle).\nA map\nπ : M → B\nbetween manifolds is a fiber bundle if there is a manifold F such that each\nb ∈ B\nhas an open neighborhood U such that there is a homeomorphism with, with\np1 : U × F → U\nprojection to the first factor.\nThe fiber bundle yields a foliation by fibers.\nIts space of leaves L is homeomorphic to B, in particular L is a Hausdorff manifold.\n\n## Coverings\nIf\nM → N\nis a [covering map](https://bliptext.com/articles/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](https://bliptext.com/articles/branched-covering), where the branch [locus](https://bliptext.com/articles/locus-mathematics) is transverse to the foliation, then the foliation can be pulled back.\n\n## Submersions\nIf\nMn → Nq, (q ≤ n)\nis a [submersion](https://bliptext.com/articles/submersion-mathematics) of manifolds, it follows from the [inverse function theorem](https://bliptext.com/articles/inverse-function-theorem) that the connected components of the fibers of the [submersion](https://bliptext.com/articles/submersion-mathematics) define a codimension q foliation of M. [Fiber bundles](https://bliptext.com/articles/fiber-bundles) are an example of this type.\nAn example of a submersion, which is not a fiber bundle, is given by\nThis submersion yields a foliation of\n[−1, 1] × R\nwhich is invariant under the\nZ\n-actions given by for\n(x, y) ∈ [−1, 1] × R\nand\nn ∈ Z\n. The induced foliations of\nZ \\ ([−1, 1] × R)\nare called the 2-dimensional Reeb foliation (of the annulus) resp.\nthe 2-dimensional nonorientable Reeb foliation (of the Möbius band).\nTheir leaf spaces are not Hausdorff.\n\n## Reeb foliations\nDefine a submersion\nwhere\n(r, θ) ∈ [0, 1] × Sn−1\nare cylindrical coordinates on the n-dimensional disk\nDn\n. This submersion yields a foliation of\nDn × R\nwhich is invariant under the\nZ\n-actions given by\nfor\n(x, y) ∈ Dn × R, z ∈ Z\n. The induced foliation of\nZ \\ (Dn × R)\nis called the n-dimensional [Reeb foliation](https://bliptext.com/articles/reeb-foliation).\nIts leaf space is not Hausdorff.\nFor\n, this gives a foliation of the solid torus which can be used to define the [Reeb foliation](https://bliptext.com/articles/reeb-foliation) of the 3-sphere by gluing two solid tori along their boundary.\nFoliations of odd-dimensional spheres\nS2n+1\nare also explicitly known.\n\n## Lie groups\nIf G is a [Lie group](https://bliptext.com/articles/lie-group), and H is a [Lie subgroup](https://bliptext.com/articles/lie-group), then G is foliated by [cosets](https://bliptext.com/articles/coset) of H. When H is [closed](https://bliptext.com/articles/closed-set) in G, the [quotient space](https://bliptext.com/articles/quotient-space-topology) G/H is a smooth ([Hausdorff](https://bliptext.com/articles/hausdorff-space)) manifold turning G into a fiber bundle with fiber H and base G/H.\nThis fiber bundle is actually [principal](https://bliptext.com/articles/principal-bundle), with structure group H.\n\n## Lie group actions\nLet G be a Lie group acting smoothly on a manifold M. If the action is a [locally free action](https://bliptext.com/articles/locally-free-action) or [free action](https://bliptext.com/articles/free-action), then the orbits of G define a foliation of M.\n\n## Linear and Kronecker foliations\nIf \\tilde{X} is a nonsingular (i.e., nowhere zero) vector field, then the local flow defined by \\tilde{X} patches together to define a foliation of dimension 1.\nIndeed, given an arbitrary point x ∈ M, the fact that \\tilde{X} is nonsingular allows one to find a coordinate neighborhood (U,x1,...,xn) about x such that\nand\nGeometrically, the flow lines of are just the level sets\nwhere all Since by convention manifolds are second countable, leaf anomalies like the \"long line\" are precluded by the second countability of M itself.\nThe difficulty can be sidestepped by requiring that \\tilde{X} be a complete field (e.g., that M be compact), hence that each leaf be a flow line.\nAn important class of 1-dimensional foliations on the torus T2 are derived from projecting constant vector fields on T2.\nA constant vector field\non ****R****2 is invariant by all translations in ****R****2, hence passes to a well-defined vector field X when projected on the torus\n. It is assumed that a ≠ 0.\nThe foliation on **R**2 produced by \\tilde{X} has as leaves the parallel straight lines of slope θ = b/a.\nThis foliation is also invariant under translations and passes to the foliation \\mathcal{F} on T2 produced by X.\nEach leaf of is of the form\nIf the slope is [rational](https://bliptext.com/articles/rational-number) then all leaves are closed curves [homeomorphic](https://bliptext.com/articles/homeomorphic) to the [circle](https://bliptext.com/articles/circle).\nIn this case, one can take a,b ∈ **Z**.\nFor fixed t ∈ **R**, the points of \\tilde{L} corresponding to values of t ∈ t0 + **Z** all project to the same point of T2; so the corresponding leaf L of \\mathcal{F} is an embedded circle in T2.\nSince L is arbitrary, \\mathcal{F} is a foliation of T2 by circles.\nIt follows rather easily that this foliation is actually a fiber bundle π : T2 → S1.\nThis is known as a linear foliation.\nWhen the slope θ = b/a is [irrational](https://bliptext.com/articles/irrational-number), the leaves are noncompact, homeomorphic to the non-compactified [real line](https://bliptext.com/articles/real-line), and [dense](https://bliptext.com/articles/dense-set) in the torus (cf [Irrational rotation](https://bliptext.com/articles/irrational-rotation)).\nThe 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.\nThus the closure to the trajectory is the entire two-dimensional torus.\nThis case is named Kronecker foliation, after [Leopold Kronecker](https://bliptext.com/articles/leopold-kronecker) and his\n**Kronecker's Density Theorem**.\nIf the real number θ is distinct from each rational multiple of π, then the set {einθ | n ∈ **Z**} is dense in the unit circle.\n!Proof\nTo see this, note first that, if a leaf \\tilde{L} of does not project one-to-one into T2, there must be a real number t ≠ 0 such that ta and tb are both integers.\nBut this would imply that b/a ∈ **Q**.\nIn order to show that each leaf L of \\mathcal{F} is dense in T2, it is enough to show that, for every v ∈ **R**2, each leaf \\tilde{L} of achieves arbitrarily small positive distances from suitable points of the coset v + **Z**2.\nA suitable translation in **R**2 allows one to assume that v = 0; so the task is reduced to showing that \\tilde{L} passes arbitrarily close to suitable points (n,m) ∈ **Z**2.\nThe line \\tilde{L} has slope-intercept equation\nSo it will suffice to find, for arbitrary η > 0, integers n and m such that\nEquivalently, c ∈ ****R**** being arbitrary, one is reduced to showing that the set {θn − m}m,n∈Z is dense in ****R****.\nThis is essentially the criterion of [Eudoxus](https://bliptext.com/articles/eudoxus-of-cnidus) that θ and 1 be incommensurable (i.e., that θ be irrational).\nA similar construction using a foliation of\nRn\nby parallel lines yields a 1-dimensional foliation of the n-torus\nRn/Zn\nassociated with the [linear flow on the torus](https://bliptext.com/articles/linear-flow-on-the-torus).\n# [The linear foliation on R2 passes to the foliation \\mathcal{F} on T2. a) the slope is rational (linear foliation); b) the slope is irrational (Kronecker foliation). | upload.wikimedia.org/wikipedia/commons/7/7e/Linear///torus.png]\n# [Irrational rotation on a 2-torus. | upload.wikimedia.org/wikipedia/commons/6/69/Irrational///Rotation///on///a///2///Torus.png]\n\n## Suspension foliations\nA flat bundle has not only its foliation by fibres but also a foliation transverse to the fibers, whose leaves are\nwhere is the canonical projection.\nThis foliation is called the suspension of the representation\nρ : π1(B) → Homeo(F)\n.\nIn particular, if\nand is a homeomorphism of F, then the suspension foliation of \\varphi is defined to be the suspension foliation of the representation\nρ : Z → Homeo(F)\ngiven by\n. Its space of leaves is\n, where\nx ~ y\nwhenever\nfor some\nn ∈ Z\n.\nThe simplest example of foliation by suspension is a manifold X of dimension q.\nLet f : X → X be a bijection.\nOne defines the suspension M = S1 ×f X as the quotient of [0,1] × X by the equivalence relation (1,x) ~ (0,f(x)).\nThen automatically M carries two foliations: \\mathcal{F}2 consisting of sets of the form F2,t = {(t,x)~ : x ∈ X} and \\mathcal{F}1 consisting of sets of the form F2,x 0 = {(t,x) : t ∈ [0,1] ,x ∈ Ox 0 }, where the orbit Ox 0 is defined as where the exponent refers to the number of times the function f is composed with itself.\nNote that Ox 0 = Of(x 0) = Of−2(x 0), etc., so the same is true for F1,x 0.\nUnderstanding the foliation \\mathcal{F}1 is equivalent to understanding the dynamics of the map f.\nIf the manifold X is already foliated, one can use the construction to increase the codimension of the foliation, as long as f maps leaves to leaves.\nThe Kronecker foliations of the 2-torus are the suspension foliations of the rotations\nRα : S1 → S1\nby angle\nα ∈ [0, 2π).\nMore specifically, if Σ = Σ2 is the two-holed torus with C1,C2 ∈ Σ the two embedded circles let \\mathcal{F} be the product foliation of the 3-manifold M = Σ × S1 with leaves Σ × {y}, y ∈ S1.\nNote that Ni = Ci × S1 is an embedded torus and that \\mathcal{F} is transverse to Ni, i = 1,2.\nLet Diff+(S1) denote the group of orientation-preserving diffeomorphisms of S1 and choose f1,f2 ∈ Diff+(S1).\nCut M apart along N1 and N2, letting N_i^{+} and N_i^{-} denote the resulting copies of Ni, i = 1,2.\nAt this point one has a manifold M' = Σ' × S1 with four boundary components The foliation \\mathcal{F} has passed to a foliation transverse to the boundary ∂M' , each leaf of which is of the form Σ' × {y}, y ∈ S1.\nThis leaf meets ∂M' in four circles If z ∈ Ci, the corresponding points in C_i^{\\pm} are denoted by z± and N_i^{-} is \"reglued\" to N_i^{+} by the identification\nSince f1 and f2 are orientation-preserving diffeomorphisms of S1, they are isotopic to the identity and the manifold obtained by this regluing operation is homeomorphic to M.\nThe leaves of, however, reassemble to produce a new foliation \\mathcal{F}(f1,f2) of M.\nIf a leaf L of \\mathcal{F}(f1,f2) contains a piece Σ' × {y0}, then\nwhere G ⊂ Diff+(S1) is the subgroup generated by {f1,f2}.\nThese copies of Σ' are attached to one another by identifications\nwhere g ranges over G.\nThe leaf is completely determined by the G-orbit of y0 ∈ S1 and can he simple or immensely complicated.\nFor instance, a leaf will be compact precisely if the corresponding G-orbit is finite.\nAs an extreme example, if G is trivial (f1 = f2 = idS1), then \\mathcal{F}(f1,f2) = \\mathcal{F}.\nIf an orbit is dense in S1, the corresponding leaf is dense in M.\nAs an example, if f1 and f2 are rotations through rationally independent multiples of 2π, every leaf will be dense.\nIn other examples, some leaf L has closure \\bar{L} that meets each factor {w} × S1 in a [Cantor set](https://bliptext.com/articles/cantor-set).\nSimilar constructions can be made on Σ × I, where I is a compact, nondegenerate interval.\nHere one takes f1,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.\nWhen one forms M' in this case, one gets a foliated manifold with corners.\nIn either case, this construction is called the suspension of a pair of diffeomorphisms and is a fertile source of interesting examples of codimension-one foliations.\n# [Suspension of 2-holed torus after cutting and re-gluing. a) Two-holed torus with the sections to be cut; b) the geometric figure after cutting with the four faces. | upload.wikimedia.org/wikipedia/commons/3/35/2-Hole///Torus-cut.png]\n\n# Foliations and integrability\nThere is a close relationship, assuming everything is [smooth](https://bliptext.com/articles/smooth-function), with [vector fields](https://bliptext.com/articles/vector-field): given a vector field X on M that is never zero, its [integral curves](https://bliptext.com/articles/integral-curve) will give a 1-dimensional foliation.\n(i.e. a codimension\nn − 1\nfoliation).\nThis observation generalises to the [Frobenius theorem](https://bliptext.com/articles/frobenius-theorem-differential-topology), saying that the [necessary and sufficient conditions](https://bliptext.com/articles/necessary-and-sufficient-conditions) for a distribution (i.e. an\nn − p\ndimensional [subbundle](https://bliptext.com/articles/subbundle) of the [tangent bundle](https://bliptext.com/articles/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](https://bliptext.com/articles/lie-bracket-of-vector-fields). One can also phrase this differently, as a question of [reduction of the structure group](https://bliptext.com/articles/reduction-of-the-structure-group) of the [tangent bundle](https://bliptext.com/articles/tangent-bundle) from\nGL(n)\nto a reducible subgroup.\nThe conditions in the Frobenius theorem appear as [integrability conditions](https://bliptext.com/articles/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.\nFor example, in the codimension 1 case, we can define the tangent bundle of the foliation as\nker(α)\n, for some (non-canonical)\nα ∈ Ω1\n(i.e. a non-zero co-vector field).\nA given α is integrable iff\nα ∧ dα = 0\neverywhere.\nThere is a global foliation theory, because topological constraints exist.\nFor example, in the [surface](https://bliptext.com/articles/surface-topology) case, an everywhere non-zero vector field can exist on an [orientable](https://bliptext.com/articles/orientable) [compact](https://bliptext.com/articles/compact-space) [surface](https://bliptext.com/articles/surface-topology) only for the [torus](https://bliptext.com/articles/torus).\nThis is a consequence of the [Poincaré–Hopf index theorem](https://bliptext.com/articles/poincar-hopf-index-theorem), which shows the [Euler characteristic](https://bliptext.com/articles/euler-characteristic) will have to be 0.\nThere are many deep connections with [contact topology](https://bliptext.com/articles/contact-topology), which is the \"opposite\" concept, requiring that the integrability condition is never satisfied.\n\n# Existence of foliations\ngave a necessary and sufficient condition for a distribution on a connected non-compact manifold to be homotopic to an integrable distribution.\nshowed that any compact manifold with a distribution has a foliation of the same dimension.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:24:54.588Z","updated_at":"2025-01-25T19:24:54.588Z","search_vector":"'foliat':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics) ([differential geometry](https://bliptext.com/articles/differential-geometry)), a **foliation** is an [equivalence relation](https://bliptext.com/articles/equivalence-relation) on an n-manifold, the [equivalence classes](https://bliptext.com/articles/equivalence-class) being connected, [injectively](https://bliptext.com/articles/injective-function) [immersed submanifolds](https://bliptext.com/articles/immersed-submanifold), all of the same dimension p, modeled on the [decomposition](https://bliptext.com/articles/manifold-decomposition) of the [real coordinate space](https://bliptext.com/articles/real-coordinate-space) ******R******n into the [cosets](https://bliptext.com/articles/coset) x + ******R******p of the standardly [embedded](https://bliptext.com/articles/embedding) [subspace](https://bliptext.com/articles/subspace-topology) ******R******p. The [equivalence classes](https://bliptext.com/articles/equivalence-class) are called the **leaves** of the **foliation**.\nIf the manifold and/or the submanifolds are required to have a [piecewise-linear](https://bliptext.com/articles/piecewise-linear-manifold), [differentiable](https://bliptext.com/articles/differentiable-manifold) (of class Cr), or [analytic](https://bliptext.com/articles/analytic-manifold) structure then one defines [piecewise-linear](https://bliptext.com/articles/piecewise-linear-manifold), [differentiable](https://bliptext.com/articles/differentiable-manifold), or [analytic](https://bliptext.com/articles/analytic-manifold) foliations, respectively.\nIn the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1 (otherwise, C0 is a topological foliation).\nThe number p (the dimension of the leaves) is called the dimension of the foliation and q = n − p is called its [codimension](https://bliptext.com/articles/codimension).\nIn some papers on [general relativity](https://bliptext.com/articles/general-relativity) by mathematical physicists, the term foliation (or **slicing**) is used to describe a situation where the relevant [Lorentz manifold](https://bliptext.com/articles/pseudo-riemannian-manifold) (a (p+1)-dimensional [spacetime](https://bliptext.com/articles/spacetime)) has been decomposed into [hypersurfaces](https://bliptext.com/articles/hypersurface) of dimension p, specified as the level sets of a real-valued [smooth function](https://bliptext.com/articles/smooth-function) ([scalar field](https://bliptext.com/articles/scalar-field)) whose [gradient](https://bliptext.com/articles/gradient) is everywhere non-zero; this [smooth function](https://bliptext.com/articles/smooth-function) is moreover usually assumed to be a **time function,** meaning that its [gradient](https://bliptext.com/articles/gradient) is everywhere [time-like](https://bliptext.com/articles/spacetime), so that its level-sets are all space-like [hypersurfaces](https://bliptext.com/articles/hypersurface).\nIn deference to standard mathematical terminology, these hypersurface are often called the leaves (or sometimes **slices**) of the foliation.\nNote 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, 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.\nFor example, while the [3-sphere](https://bliptext.com/articles/3-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.\n# [2-dimensional section of Reeb foliation | upload.wikimedia.org/wikipedia/commons/c/cb/Reebfoliation-ring-2d-2.svg]\n# [3-dimensional model of Reeb foliation | upload.wikimedia.org/wikipedia/commons/c/cc/Reeb///foliation///half-torus///POV-Ray.png]\n\n# Foliated charts and atlases\nIn order to give a more precise definition of foliation, it is necessary to define some auxiliary elements.\nA rectangular [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) in **R**n is an [open](https://bliptext.com/articles/open-set) [subset](https://bliptext.com/articles/subset) of the form B = J1 × ⋅⋅⋅ × Jn, where Ji is a (possibly unbounded) relatively [open](https://bliptext.com/articles/open-set) interval in the ith coordinate axis.\nIf J1 is of the form (a,0], it is said that B has [boundary](https://bliptext.com/articles/boundary-topology) In the following definition, coordinate charts are considered that have values in ****R****p × ****R****q, allowing the possibility of manifolds with [boundary](https://bliptext.com/articles/boundary-topology) and ([convex](https://bliptext.com/articles/convex-set)) corners.\nA foliated chart on the n-manifold M of codimension q is a pair (U,φ), where U ⊆ M is open and is a [diffeomorphism](https://bliptext.com/articles/diffeomorphism), being a rectangular neighborhood in ****R****q and B_{\\tau} a rectangular neighborhood in ****R****p. The set Py = φ−1(Bτ × {y}), where, is called a plaque of this foliated chart.\nFor each x ∈ Bτ, the set Sx = φ−1({x} × ) is called a [transversal](https://bliptext.com/articles/transversal-geometry) of the foliated chart.\nThe set ∂τU = φ−1(Bτ × (∂)) is called the tangential boundary of U and = φ−1((∂Bτ) × ) is called the transverse boundary of U.\nThe foliated chart is the basic model for all foliations, the plaques being the leaves.\nThe notation Bτ is read as \"B-tangential\" and as \"B-transverse\".\nThere are also various possibilities.\nIf both and Bτ have empty boundary, the foliated chart models codimension-q foliations of n-manifolds without boundary.\nIf 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.\nSpecifically, if ∂ ≠ ∅ = ∂Bτ, then ∂U = ∂τU is a union of plaques and the foliation by plaques is tangent to the boundary.\nIf ∂Bτ ≠ ∅ = ∂, then ∂U = is a union of transversals and the foliation is transverse to the boundary.\nFinally, if ∂ ≠ ∅ ≠ ∂Bτ, this is a model of a foliated manifold with a corner separating the tangential boundary from the transverse boundary.\nA foliated [atlas](https://bliptext.com/articles/atlas-topology) of codimension q and class Cr (0 ≤ r ≤ ∞) on the n-manifold M is a Cr-[atlas](https://bliptext.com/articles/atlas-topology) of foliated charts of codimension q which are coherently foliated in the sense that, whenever P and Q are plaques in distinct charts of \\mathcal{U}, then P ∩ Q is open both in P and Q.\nA useful way to reformulate the notion of coherently foliated charts is to write for w ∈ Uα ∩ Uβ\nThe notation (Uα,φα) is often written (Uα,xα,yα), with\nOn 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.\nEquivalently, 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\nis independent of xβ.\nThe main use of foliated atlases is to link their overlapping plaques to form the leaves of a foliation.\nFor this and other purposes, the general definition of foliated atlas above is a bit clumsy.\nOne problem is that a plaque of (Uα,φα) can meet multiple plaques of (Uβ,φβ).\nIt can even happen that a plaque of one chart meets infinitely many plaques of another chart.\nHowever, no generality is lost in assuming the situation to be much more regular as shown below.\nTwo foliated atlases \\mathcal{U} and \\mathcal{V} on M of the same codimension and [smoothness](https://bliptext.com/articles/smoothness) class Cr are coherent if is a foliated Cr-atlas.\nCoherence of foliated atlases is an equivalence relation.\n!Proof\n[Reflexivity](https://bliptext.com/articles/reflexive-relation) and [symmetry](https://bliptext.com/articles/symmetric-relation) are immediate.\nTo prove [transitivity](https://bliptext.com/articles/transitive-relation) let and.\nLet (Uα,xα,yα) ∈ \\mathcal{U} and (Wλ,xλ,yλ) ∈ \\mathcal{W} and suppose that there is a point w ∈ Uα ∩ Wλ.\nChoose (Vδ,xδ,yδ) ∈ \\mathcal{V} such that w ∈ Vδ.\nBy the above remarks, there is a neighborhood N of w in Uα ∩ Vδ ∩ Wλ such that\nand hence\nSince w ∈ Uα ∩ Wλ is arbitrary, it can be concluded that yα(xλ,yλ) is locally independent of xλ.\nIt is thus proven that, hence that coherence is transitive.\nPlaques and transversals defined above on open sets are also open.\nBut one can speak also of closed plaques and transversals.\nNamely, if (U,φ) and (W,ψ) are foliated charts such that (the [closure](https://bliptext.com/articles/closure-mathematics) of U) is a subset of W and φ = ψ|U then, if it can be seen that, written , carries diffeomorphically onto\nA foliated atlas is said to be regular if\nBy 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\nSimilar assertions hold also for open charts (without the overlines).\nThe transverse coordinate map yα can be viewed as a [submersion](https://bliptext.com/articles/submersion-mathematics)\nand the formulas yα = yα(yβ) can be viewed as [diffeomorphisms](https://bliptext.com/articles/diffeomorphisms)\nThese satisfy the [cocycle conditions](https://bliptext.com/articles/cocycle-condition).\nThat is, on yδ(Uα ∩ Uβ ∩ Uδ),\nand, in particular,\nUsing the above definitions for coherence and regularity it can be proven that every foliated atlas has a coherent [refinement](https://bliptext.com/articles/refinement-topology) that is regular.\n!Proof\nFix a metric on M and a foliated atlas Passing to a [subcover](https://bliptext.com/articles/cover-topology), if necessary, one can assume that is finite.\nLet ε > 0 be a [Lebesgue number](https://bliptext.com/articles/lebesgue-number-lemma) for That is, any subset X ⊆ M of diameter 0 so small that εl k ≥ 0, if the corresponding coherence class of foliated atlases contains a regular foliated atlas {Uα,xα,yα}α∈A such that the change of coordinate formula\nis of class Ck, but xα is of class Cr in the coordinates xβ and its mixed xβ partials of orders ≤ r are Ck in the coordinates (xβ,yβ).\nThe above definition suggests the more general concept of a foliated space or abstract lamination.\nOne relaxes the condition that the transversals be open, relatively compact subsets of **R**q, allowing the transverse coordinates yα to take their values in some more general topological space Z.\nThe plaques are still open, relatively compact subsets of **R**p, the change of transverse coordinate formula yα(yβ) is continuous and xα(xβ,yβ) is of class Cr in the coordinates xβ and its mixed xβ partials of orders ≤ r are continuous in the coordinates (xβ,yβ).\nOne usually requires M and Z to be locally compact, [second countable](https://bliptext.com/articles/second-countable) and metrizable.\nThis may seem like a rather wild generalization, but there are contexts in which it is useful.\n# [Decomposition through the coordinates function x : U→Rn. | upload.wikimedia.org/wikipedia/commons/f/fe/Foliation///definition.svg]\n\n# Holonomy\nLet (M, \\mathcal{F}) be a foliated manifold.\nIf L is a leaf of \\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.\nIntuitively, an inhabitant of the leaf walks along the path s, keeping an eye on all of the nearby leaves.\nAs 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.\nIf 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.\nThis behavior, when appropriately formalized, is called the holonomy of the foliation.\nHolonomy 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.\n\n## Foliated bundles\nThe easiest case of holonomy to understand is the total holonomy of a foliated bundle.\nThis is a generalization of the notion of a [Poincaré map](https://bliptext.com/articles/poincar-map).\nThe term [\"first return (recurrence) map\"](https://bliptext.com/articles/poincar-map) comes from the theory of dynamical systems.\nLet Φt be a nonsingular Cr flow (r ≥ 1) on the compact n-manifold M.\nIn applications, one can imagine that M is a [cyclotron](https://bliptext.com/articles/cyclotron) or some closed loop with fluid flow.\nIf M has a boundary, the flow is assumed to be tangent to the boundary.\nThe flow generates a 1-dimensional foliation \\mathcal{F}.\nIf one remembers the positive direction of flow, but otherwise forgets the parametrization (shape of trajectory, velocity, etc.), the underlying foliation \\mathcal{F} is said to be oriented.\nSuppose that the flow admits a global cross section N.\nThat is, N is a compact, properly embedded, Cr submanifold of M of dimension n – 1, the foliation \\mathcal{F} is transverse to N, and every flow line meets N.\nBecause the dimensions of N and of the leaves are complementary, the transversality condition is that\nLet y ∈ N and consider the ω-[limit set](https://bliptext.com/articles/limit-set) ω(y) of all accumulation points in M of all sequences, where tk goes to infinity.\nIt can be shown that ω(y) is compact, nonempty, and a union of flow lines.\nIf there is a value t* ∈ **R** such that Φt*(z) ∈ N and it follows that\nSince N is compact and \\mathcal{F} is transverse to N, it follows that the set {t > 0 | Φt(y) ∈ N} is a monotonically increasing sequence that diverges to infinity.\nAs y ∈ N varies, let τ(y) = τ1(y), defining in this way a positive function τ ∈ Cr(N) (the first return time) such that, for arbitrary y ∈ N, Φt(y) ∉ N, 0 0, integers n and m such that\nEquivalently, c ∈ ****R**** being arbitrary, one is reduced to showing that the set {θn − m}m,n∈Z is dense in ****R****.\nThis is essentially the criterion of [Eudoxus](https://bliptext.com/articles/eudoxus-of-cnidus) that θ and 1 be incommensurable (i.e., that θ be irrational).\nA similar construction using a foliation of\nRn\nby parallel lines yields a 1-dimensional foliation of the n-torus\nRn/Zn\nassociated with the [linear flow on the torus](https://bliptext.com/articles/linear-flow-on-the-torus).\n# [The linear foliation on R2 passes to the foliation \\mathcal{F} on T2. a) the slope is rational (linear foliation); b) the slope is irrational (Kronecker foliation). | upload.wikimedia.org/wikipedia/commons/7/7e/Linear///torus.png]\n# [Irrational rotation on a 2-torus. | upload.wikimedia.org/wikipedia/commons/6/69/Irrational///Rotation///on///a///2///Torus.png]\n\n## Suspension foliations\nA flat bundle has not only its foliation by fibres but also a foliation transverse to the fibers, whose leaves are\nwhere is the canonical projection.\nThis foliation is called the suspension of the representation\nρ : π1(B) → Homeo(F)\n.\nIn particular, if\nand is a homeomorphism of F, then the suspension foliation of \\varphi is defined to be the suspension foliation of the representation\nρ : Z → Homeo(F)\ngiven by\n. Its space of leaves is\n, where\nx ~ y\nwhenever\nfor some\nn ∈ Z\n.\nThe simplest example of foliation by suspension is a manifold X of dimension q.\nLet f : X → X be a bijection.\nOne defines the suspension M = S1 ×f X as the quotient of [0,1] × X by the equivalence relation (1,x) ~ (0,f(x)).\nThen automatically M carries two foliations: \\mathcal{F}2 consisting of sets of the form F2,t = {(t,x)~ : x ∈ X} and \\mathcal{F}1 consisting of sets of the form F2,x 0 = {(t,x) : t ∈ [0,1] ,x ∈ Ox 0 }, where the orbit Ox 0 is defined as where the exponent refers to the number of times the function f is composed with itself.\nNote that Ox 0 = Of(x 0) = Of−2(x 0), etc., so the same is true for F1,x 0.\nUnderstanding the foliation \\mathcal{F}1 is equivalent to understanding the dynamics of the map f.\nIf the manifold X is already foliated, one can use the construction to increase the codimension of the foliation, as long as f maps leaves to leaves.\nThe Kronecker foliations of the 2-torus are the suspension foliations of the rotations\nRα : S1 → S1\nby angle\nα ∈ [0, 2π).\nMore specifically, if Σ = Σ2 is the two-holed torus with C1,C2 ∈ Σ the two embedded circles let \\mathcal{F} be the product foliation of the 3-manifold M = Σ × S1 with leaves Σ × {y}, y ∈ S1.\nNote that Ni = Ci × S1 is an embedded torus and that \\mathcal{F} is transverse to Ni, i = 1,2.\nLet Diff+(S1) denote the group of orientation-preserving diffeomorphisms of S1 and choose f1,f2 ∈ Diff+(S1).\nCut M apart along N1 and N2, letting N_i^{+} and N_i^{-} denote the resulting copies of Ni, i = 1,2.\nAt this point one has a manifold M' = Σ' × S1 with four boundary components The foliation \\mathcal{F} has passed to a foliation transverse to the boundary ∂M' , each leaf of which is of the form Σ' × {y}, y ∈ S1.\nThis leaf meets ∂M' in four circles If z ∈ Ci, the corresponding points in C_i^{\\pm} are denoted by z± and N_i^{-} is \"reglued\" to N_i^{+} by the identification\nSince f1 and f2 are orientation-preserving diffeomorphisms of S1, they are isotopic to the identity and the manifold obtained by this regluing operation is homeomorphic to M.\nThe leaves of, however, reassemble to produce a new foliation \\mathcal{F}(f1,f2) of M.\nIf a leaf L of \\mathcal{F}(f1,f2) contains a piece Σ' × {y0}, then\nwhere G ⊂ Diff+(S1) is the subgroup generated by {f1,f2}.\nThese copies of Σ' are attached to one another by identifications\nwhere g ranges over G.\nThe leaf is completely determined by the G-orbit of y0 ∈ S1 and can he simple or immensely complicated.\nFor instance, a leaf will be compact precisely if the corresponding G-orbit is finite.\nAs an extreme example, if G is trivial (f1 = f2 = idS1), then \\mathcal{F}(f1,f2) = \\mathcal{F}.\nIf an orbit is dense in S1, the corresponding leaf is dense in M.\nAs an example, if f1 and f2 are rotations through rationally independent multiples of 2π, every leaf will be dense.\nIn other examples, some leaf L has closure \\bar{L} that meets each factor {w} × S1 in a [Cantor set](https://bliptext.com/articles/cantor-set).\nSimilar constructions can be made on Σ × I, where I is a compact, nondegenerate interval.\nHere one takes f1,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.\nWhen one forms M' in this case, one gets a foliated manifold with corners.\nIn either case, this construction is called the suspension of a pair of diffeomorphisms and is a fertile source of interesting examples of codimension-one foliations.\n# [Suspension of 2-holed torus after cutting and re-gluing. a) Two-holed torus with the sections to be cut; b) the geometric figure after cutting with the four faces. | upload.wikimedia.org/wikipedia/commons/3/35/2-Hole///Torus-cut.png]\n\n# Foliations and integrability\nThere is a close relationship, assuming everything is [smooth](https://bliptext.com/articles/smooth-function), with [vector fields](https://bliptext.com/articles/vector-field): given a vector field X on M that is never zero, its [integral curves](https://bliptext.com/articles/integral-curve) will give a 1-dimensional foliation.\n(i.e. a codimension\nn − 1\nfoliation).\nThis observation generalises to the [Frobenius theorem](https://bliptext.com/articles/frobenius-theorem-differential-topology), saying that the [necessary and sufficient conditions](https://bliptext.com/articles/necessary-and-sufficient-conditions) for a distribution (i.e. an\nn − p\ndimensional [subbundle](https://bliptext.com/articles/subbundle) of the [[tangent bundle](https://bliptext.com/articles/tangent-bundle)](https://bliptext.com/articles/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](https://bliptext.com/articles/lie-bracket-of-vector-fields). One can also phrase this differently, as a question of [reduction of the structure group](https://bliptext.com/articles/reduction-of-the-structure-group) of the [[tangent bundle](https://bliptext.com/articles/tangent-bundle)](https://bliptext.com/articles/tangent-bundle) from\nGL(n)\nto a reducible subgroup.\nThe conditions in the Frobenius theorem appear as [integrability conditions](https://bliptext.com/articles/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.\nFor example, in the codimension 1 case, we can define the tangent bundle of the foliation as\nker(α)\n, for some (non-canonical)\nα ∈ Ω1\n(i.e. a non-zero co-vector field).\nA given α is integrable iff\nα ∧ dα = 0\neverywhere.\nThere is a global foliation theory, because topological constraints exist.\nFor example, in the [surface](https://bliptext.com/articles/surface-topology) case, an everywhere non-zero vector field can exist on an [orientable](https://bliptext.com/articles/orientable) [compact](https://bliptext.com/articles/compact-space) [surface](https://bliptext.com/articles/surface-topology) only for the [torus](https://bliptext.com/articles/torus).\nThis is a consequence of the [Poincaré–Hopf index theorem](https://bliptext.com/articles/poincar-hopf-index-theorem), which shows the [Euler characteristic](https://bliptext.com/articles/euler-characteristic) will have to be 0.\nThere are many deep connections with [contact topology](https://bliptext.com/articles/contact-topology), which is the \"opposite\" concept, requiring that the integrability condition is never satisfied.\n\n# Existence of foliations\ngave a necessary and sufficient condition for a distribution on a connected non-compact manifold to be homotopic to an integrable distribution.\nshowed that any compact manifold with a distribution has a foliation of the same dimension.","revisionCount":0},"hasMore":false,"nextCursor":null}

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