Foliation
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.
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.
[3] Note that while this situation does constitute a codimension-1 foliation in the standard mathematical sense, examples of this type are actually globally trivial; while the leaves of a (mathematical) codimension-1 foliation are always locally the level sets of a function, they generally cannot be expressed this way globally,[4][5] as a leaf may pass through a local-trivializing chart infinitely many times, and the holonomy around a leaf may also obstruct the existence of a globally-consistent defining functions for the leaves.
For example, while the 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.
For each x ∈ M, choose j such that x ∈ Wj and choose a foliated chart (Ux, φx) such that Suppose that Ux ⊂ Wk, k ≠ j, and write ψk = (xk,yk) as usual, where yk : Wk → Rq is the transverse coordinate map.
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 Rn containing x such that every leaf, Ma, meets U in either the empty set or a countable collection of subspaces whose images under
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.
of an n-dimensional manifold M that is a covered by charts Ui together with maps such that for overlapping pairs Ui, Uj the transition functions φij : Rn → Rn defined by take the form where x denotes the first q = n − p coordinates, and y denotes the last p co-ordinates.
As 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.
If one remembers the positive direction of flow, but otherwise forgets the parametrization (shape of trajectory, velocity, etc.
Suppose that the flow admits a global cross section N. That is, N is a compact, properly embedded, Cr submanifold of M of dimension n – 1, the foliation
It is possible to reparametrize the flow Φt, keeping it nonsingular, of class Cr, and not reversing its direction, so that τ ≡ 1.
The assumption that there is a cross section N to the flow is very restrictive, implying that M is the total space of a fiber bundle over S1.
The 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. The diffeomorphism fk ∈ Diffr(N) is also denoted by hs and is called the total holonomy of the loop s. Since this depends only on [s], this is a definition of a homomorphism called the total holonomy homomorphism for the foliated bundle.
Finally, for each x ∈ B, assume that there is a connected, open neighborhood U ⊆ B of x and a local trivialization where φ is a Cr diffeomorphism (a homeomorphism, if r = 0) that carries
A simple version of the problem is a foliation of R2, transverse to the fibration but with infinitely many leaves missing the y-axis.
One 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.
Since B is assumed to support a C∞ structure, according to the Whitehead theorem one can fix a Riemannian metric on B and choose the atlas
by setting Consider an n-dimensional space, foliated as a product by subspaces consisting of points whose first n − p coordinates are constant.
The analogy is seen directly in three dimensions, by taking n = 3 and p = 2: the 2-dimensional leaves of a book are enumerated by a (1-dimensional) page number.
is nonsingular allows one to find a coordinate neighborhood (U,x1,...,xn) about x such that and Geometrically, the flow lines of
An important class of 1-dimensional foliations on the torus T2 are derived from projecting constant vector fields on T2.
If the real number θ is distinct from each rational multiple of π, then the set {einθ | n ∈ Z} is dense in the unit circle.
1 consisting of sets of the form F2,x0 = {(t,x) : t ∈ [0,1] ,x ∈ Ox0}, where the orbit Ox0 is defined as where the exponent refers to the number of times the function f is composed with itself.
by the identification Since 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. The leaves of
These copies of Σ' are attached to one another by identifications where g ranges over G. The leaf is completely determined by the G-orbit of y0 ∈ S1 and can he simple or immensely complicated.
In 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.
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.
There are many deep connections with contact topology, which is the "opposite" concept, requiring that the integrability condition is never satisfied.
