Foliation

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.

Definition

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

:$phi\_i:U\_i\; o\; R^n$

such that on the overlaps $U\_i\; cap\; U\_j$ the transition functions $varphi\_\{ij\}:mathbb\{R\}^n\; omathbb\{R\}^n$ 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,:$varphi\_\{ij\}^1:mathbb\{R\}^\{n-p\}\; omathbb\{R\}^\{n-p\}$ and :$varphi\_\{ij\}^2:mathbb\{R\}^n\; omathbb\{R\}^\{p\}$. 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.

If we shrink the chart $U\_i$ it can be written in the form $U\_\{ix\}\; imes\; U\_\{iy\}$ where $U\_\{ix\}subsetmathbb\{R\}^\{n-p\}$ and $U\_\{iy\}subsetmathbb\{R\}^p$ and $U\_\{iy\}$ is isomorphic to the plaques and the points of $U\_\{ix\}$ parametrize the plaques in $U\_i$. If we pick a $y\_0in\; U\_\{iy\}$, $U\_\{ix\}\; imes\{y\_0\}$ 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

:$mathbb\{R\}^n=mathbb\{R\}^\{n-p\}\; imes\; mathbb\{R\}^\{p\}$

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 $M\; o\; N$ 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.

ubmersions

If $M^n\; o\; N^q$ (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 $M$.

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 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 $GL(n)$ 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.

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.

ee also

*G-structure
*Classifying space for foliations
*Reeb foliation
*Taut foliation

References

*Lawson, H. Blaine, [http://www.ams.org/bull/1974-80-03/S0002-9904-1974-13432-4/S0002-9904-1974-13432-4.pdf "Foliations"]
*I.Moerdijk, J. Mrčun: Introduction to Foliations and Lie groupoids, Cambridge University Press 2003, ISBN 0521831970 (with proofs)