foliation: Definition from Answers.com

For other uses, see Foliation (disambiguation).

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.

1 Definition
2 Examples
3 Foliations and integrability
4 Existence of foliations
5 See also
6 References

Definition

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

$\phi_i:U_i \to \R^n$

such that on the overlaps $U_i \cap U_j$ the transition functions $\varphi_{ij}:\mathbb{R}^n\to\mathbb{R}^n$ 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 $n - p$ co-ordinates, and $y$ denotes the last p co-ordinates. That is,

$\varphi_{ij}^1:\mathbb{R}^{n-p}\to\mathbb{R}^{n-p}$

and

$\varphi_{ij}^2:\mathbb{R}^n\to\mathbb{R}^{p}.$

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 $n - p$ 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 a $n$ -manifold $M$ may be thought of as simply a collection $M a$ of pairwise-disjoint, connected $p$ -dimensional sub-manifolds (the leaves of the foliation) of $M$ , such that for every point $x$ in $M$ , there is a chart $(U,φ)$ with $U$ homeomorphic to $\mathbb{R}^{n}$ containing $x$ such that for every leaf $M a$ , $M a$ meets $U$ in either the empty set or a countable collection of subspaces whose preimages in $U$ are $p$ -dimensional affine subspaces whose last $n - p$ coordinates are constant.

If we shrink the chart $U i$ it can be written in the form $U_{ix}\times U_{iy}$ where $U_{ix}\subset\mathbb{R}^{n-p}$ and $U_{iy}\subset\mathbb{R}^p$ and $U i y$ is isomorphic to the plaques and the points of $U i x$ parametrize the plaques in $U i$ . If we pick a $y_0\in U_{iy}$ , $U_{ix}\times\{y_0\}$ 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 there might not exist global transversal sections of the foliation.

Examples

Flat space

Consider an $n$ -dimensional space, foliated as a product by subspaces consisting of points whose first $n - p$ co-ordinates are constant. This can be covered with a single chart. The statement is essentially that

$\mathbb{R}^n=\mathbb{R}^{n-p}\times \mathbb{R}^{p}$

with the leaves or plaques $\mathbb{R}^{n-p}$ being enumerated by $\mathbb{R}^{p}$ . The analogy is seen directly in three dimensions, by taking $n = 3$ and $p = 1$ : the two-dimensional leaves of a book are enumerated by a (one-dimensional) page number.

Covers

If $M \to N$ 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 $M^n \to N^q$ (where $q \leq n$ ) 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.

Lie groups

If $G$ is a Lie group, and $H$ is a subgroup obtained by exponentiating a closed subalgebra of the Lie algebra of $G$ , then $G$ is foliated by cosets of $H$ .

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. A similar construction using a foliation of Rⁿ by parallel lines yields a one-dimensional foliation of the n-torus Rⁿ/Zⁿ associated with the linear flow on the torus.

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, are 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 $G L (n)$ 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.

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.

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.

References

Anosov, D.V. (2001), "Foliation", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Haefliger, André (1970), "Feuilletages sur les variétés ouvertes", Topology. An International Journal of Mathematics 9: 183–194, doi:10.1016/0040-9383(70)90040-6, MR 0263104, ISSN 0040-9383
Lawson, H. Blaine (1974), "Foliations", Bulletin of the American Mathematical Society 80: 369–418, doi:10.1090/S0002-9904-1974-13432-4, MR 0343289, ISSN 0002-9904, http://www.ams.org/bull/1974-80-03/S0002-9904-1974-13432-4/
Moerdijk, Ieke; Mrčun, J. (2003), Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, MR 2012261, ISBN 978-0-521-83197-0
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, MR 0370619, ISSN 0010-2571, http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=213742
Thurston, William P. (1976), "Existence of codimension-one foliations", Annals of Mathematics. Second Series 104 (2): 249–268, MR 0425985, ISSN 0003-486X, http://www.jstor.org/stable/1971047

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