Foliation

For other uses, see Foliation (disambiguation).

In mathematics, a foliation is a geometric device used to study manifolds. Informally speaking, a foliation is a kind of "clothing" worn on a manifold, cut from a striped fabric. On each sufficiently small piece of the manifold, these stripes give the manifold a local product structure. This product structure does not have to be consistent outside local patches (i.e., well-defined globally): a stripe followed around long enough might return to a different, nearby stripe.

1 Definition
2 Examples
3 Foliations and integrability
4 See also
5 References

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.

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.

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.

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.

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 $H$ .

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 $M$ .

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 a theorem of Ferdinand Georg Frobenius (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.

Lawson, H. Blaine, "Foliations"
I.Moerdijk, J. Mrčun: Introduction to Foliations and Lie groupoids, Cambridge University Press 2003, ISBN 0521831970 (with proofs)

Foliation

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