Foliation

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

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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

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stripe.

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More formally, a codimension p foliation F of an n-dimensional manifold

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M is a covering by charts U_i together with maps

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:phi_i:U_i o R^n

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such that on the overlaps U_i cap U_j the transition functions arphi_{ij} defined by

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:arphi_{ij} =phi_j phi_i^{-1}

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take the form

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:arphi_{ij}(x,y) = (arphi_{ij}^1(x),arphi_{ij}^2(x,y))

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

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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 submanifolds called the leaves of the foliation.

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Example: n-dimensional space, foliated as a product by subspaces consisting of points whose

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first n-p co-ordinates are constant. This can be covered with a single chart.

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Example: If M o N is a covering between manifolds, and F is a foliation

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on N, then it pulls back to a foliation on M. More generally, if the map is merely

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a branched covering, where the branch locus is transverse to the foliation, then the foliation can be pulled back.

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Example: 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.

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There is a close relationship, assuming everything is smooth, with vector fields: given a vector field X

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on M that is never zero, its integral curves will give a 1-dimensional foliation. (i.e. a codimension n-1 foliation).

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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.

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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.

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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 Poincar?-Hopf index theorem, which shows the Euler characteristic will have to be 0.

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