Foliation

In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension.

Definition

More formally, a dimension p foliation F of an n-dimensional manifold M is a covering by charts U_i together with maps

\( \phi_i:U_i \to \mathbf{R}^n \)

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

\( \varphi_{ij} =\phi_j \phi_i^{-1} \)

take the form

\( \varphi_{ij}(x,y) = (\varphi_{ij}^1(x),\varphi_{ij}^2(x,y)) \)

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

\( \begin{align} \varphi_{ij}^1: &\mathbf{R}^{n-p}\to\mathbf{R}^{n-p} \\ \varphi_{ij}^2: &\mathbf{R}^n\to\mathbf{R}^{p} \end{align} \)

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

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

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

Examples
Flat space

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 Rⁿ = R^n−p × R^p with the leaves or plaques R^p being enumerated by . The analogy is seen directly in three dimensions, by taking R^n-pn = 3 and p = 2: the 2-dimensional leaves of a book are enumerated by a (1-dimensional) page number.

Bundles

A rather trivial example of foliations are products M = B × F, foliated by the leaves F_b = {b} × F, b ∈ B. (Another foliation of M is given by B_f = B × { f } ,  f  ∈ F.)

A more general class are flat G-bundles with G = Homeo(F) for a manifold F. Given a representation ρ : π1(B) → Homeo(F), the flat Homeo(F)-bundle with monodromy ρ is given by \( M=\left(\widetilde{B}\times F\right)/\pi_1B \), where π1(B) acts on the universal cover \( \widetilde{B \)} by deck transformations and on F by means of the representation ρ.

Flat bundles fit into the framework of fiber bundles. A map π : M → B 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 \( \phi:\pi^{-1}(U)\to U\times F \) with \( \pi=p_1\phi \), with p1 : U × F → U projection to the first factor. The fiber bundle yields a foliation by fibers \( F_b:=\pi^{-1}(\{b\}), b\in B \). Its space of leaves L is homeomorphic to B, in particular L is a Hausdorff manifold.

Covers

If M → N is a covering 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.

Submersions

If Mⁿ → N^q, (q ≤ n) is a submersion of manifolds, it follows from the inverse function theorem that the connected components of the fibers of the submersion define a codimension q foliation of M. Fiber bundles are an example of this type.

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

\( \begin{cases} f:[-1,1]\times \mathbf{R}\to \mathbf{R} \\f(x,y)=\left(x^2-1\right)e^y\end{cases} \)

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

\( z(x, y)= (x,y+z ), \quad \text{or} \quad z(x, y)=\left((-1)^zx, y\right) \)

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

Reeb foliations

Define a submersion

\( \begin{cases} f:D^{n}\times \mathbf{R}\to \mathbf{R} \\ f(r,\theta,t):=\left(r^2-1\right)e^t\end{cases} \)

where (r, θ) ∈ [0, 1] × S ^n-1 are cylindrical coordinates on the n-dimensional disk Dⁿ. This submersion yields a foliation of Dⁿ × R which is invariant under the Z-actions given by

\( z(x,y)=(x,y+z) \)

for (x, y) ∈ Dⁿ × R, z ∈ Z. The induced foliation of Z\(Dⁿ × R) is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.

For n = 2, this gives a foliation of the solid torus which can be used to define the Reeb foliation of the 3-sphere by gluing two solid tori along their boundary. Foliations of odd-dimensional spheres S²ⁿ⁺¹ are also explicitly known.^[1]

Lie groups

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.
Lie group actions

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.

Kronecker foliation

The set of lines on the torus T = R²/Z² with the same slope θ forms a foliation. The leaves are obtained by projecting straight lines of slope θ in the plane R² onto the torus. If the slope is rational then all leaves are closed curves homeomorphic to the circle, while if it is irrational, the leaves are noncompact, homeomorphic to the real line, and dense in the torus (cf Irrational rotation). The irrational case is known as the Kronecker foliation, after Leopold Kronecker. A similar construction using a foliation of Rⁿ by parallel lines yields a 1-dimensional foliation of the n-torus Rⁿ/Zⁿ associated with the linear flow on the torus.

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

\( L_f:= \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}, \quad \mbox{ for }f\in F, \)

where \( p:\widetilde{B}\times F\to M \) is the canonical projection. This foliation is called the suspension of the representation ρ : π1(B) → Homeo(F).

In particular, if B = S1 and \( \phi:F\to F \) is a homeomorphism of F, then the suspension foliation of \phi is defined to be the suspension foliation of the representation ρ : Z → Homeo(F) given by ρ(z) = Φz. Its space of leaves is L = F/~, where x ~ y whenever y = Φn(x) for some n ∈ Z.

The Kronecker foliations of the 2-torus are the suspension foliations of the rotations Rα : S1 → S1 by angle α ∈ [0, 2π).

Foliations and integrability

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

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

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

There is a global foliation theory, because topological constraints exist. For example in the surface case, an everywhere non-zero vector field can exist on an orientable compact surface only for the torus. This is a consequence of the Poincaré–Hopf index theorem, which shows the Euler characteristic will have to be 0. There are many deep connections with contact topology, which is the "opposite" concept.

Existence of foliations

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

G-structure
Classifying space for foliations
Haefliger structure, a generalization of foliations closed under taking pullbacks.
Lamination
Reeb foliation of the 3-sphere.
Taut foliation

References

Durfee: Foliations of Odd-Dimensional Spheres. Annals of Mathematics, Second Series, Vol. 96, No. 2 (Sep., 1972), pp. 407-411.

General

Anosov, D.V. (2001), "Foliation", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Haefliger, André (1970), "Feuilletages sur les variétés ouvertes", Topology. an International Journal of Mathematics 9 (2): 183–194, doi:10.1016/0040-9383(70)90040-6, ISSN 0040-9383, MR 0263104
Lawson, H. Blaine (1974), "Foliations", Bulletin of the American Mathematical Society 80 (3): 369–418, doi:10.1090/S0002-9904-1974-13432-4, ISSN 0002-9904, MR 0343289
Moerdijk, Ieke; Mrčun, J. (2003), Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, ISBN 978-0-521-83197-0, MR 2012261
Reeb, Georges (1952), Sur certaines propriétés topologiques des variétés feuilletées, Actualités Sci. Ind., no. 1183, Hermann & Cie., Paris, MR 0055692
Thurston, William (1974), "The theory of foliations of codimension greater than one", Commentarii Mathematici Helvetici 49: 214–231, doi:10.1007/BF02566730, ISSN 0010-2571, MR 0370619
Thurston, William P. (1976), "Existence of codimension-one foliations", Annals of Mathematics. Second Series (Annals of Mathematics) 104 (2): 249–268, doi:10.2307/1971047, ISSN 0003-486X, JSTOR 1971047, MR 0425985

External links

Foliations at the Manifold Atlas