# Foliation

In mathematics, informally speaking, a foliation is a kind of clothing worn on a manifold, cut from a stripy 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.

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

[itex]\phi_i:U_i \to \R^n[itex]

such that on the overlaps [itex]U_i \cap U_j[itex] the transition functions [itex]\varphi_{ij}[itex] defined by

[itex]\varphi_{ij} =\phi_j \phi_i^{-1}[itex]

take the form

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

where [itex]x[itex] denotes the first [itex]n-p[itex] co-ordinates, and [itex]y[itex] denotes the last p co-ordinates. In the chart [itex]U_i[itex], the stripes [itex]x=[itex]constant match up with the stripes on other charts [itex]U_j[itex].

Technically, these stripes are called plaques of the foliation. In each chart, the plaques are [itex]n-p[itex] dimensional submanifolds. These submanifolds piece together from chart to chart to form maximal connected submanifolds called the leaves of the foliation.

Example: [itex]n[itex]-dimensional space, foliated as a product by subspaces consisting of points whose first [itex]n-p[itex] co-ordinates are constant. This can be covered with a single chart.

Example: If [itex]M \to N [itex] is a covering between manifolds, and [itex]F[itex] is a foliation on [itex]N[itex], then it pulls back to a foliation on [itex]M[itex]. 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.

Example: If [itex]G[itex] is a Lie group, and [itex]H[itex] is a subgroup obtained by exponentiating a closed subalgebra of the Lie algebra of [itex]G[itex], then [itex]G[itex] is foliated by cosets of [itex]H[itex].

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

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