Foliation
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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 <math>p<math> foliation <math>F<math> of an <math>n<math>-dimensional manifold <math>M<math> is a covering by charts <math>U_i<math> together with maps
- <math>\phi_i:U_i \to \R^n<math>
such that on the overlaps <math>U_i \cap U_j<math> the transition functions <math>\varphi_{ij}<math> defined by
- <math>\varphi_{ij} =\phi_j \phi_i^{-1}<math>
take the form
- <math>\varphi_{ij}(x,y) = (\varphi_{ij}^1(x),\varphi_{ij}^2(x,y))<math>
where <math>x<math> denotes the first <math>n-p<math> co-ordinates, and <math>y<math> denotes the last p co-ordinates. In the chart <math>U_i<math>, the stripes <math>x=<math>constant match up with the stripes on other charts <math>U_j<math>.
Technically, these stripes are called plaques of the foliation. In each chart, the plaques are <math>n-p<math> dimensional submanifolds. These submanifolds piece together from chart to chart to form maximal connected submanifolds called the leaves of the foliation.
Example: <math>n<math>-dimensional space, foliated as a product by subspaces consisting of points whose first <math>n-p<math> co-ordinates are constant. This can be covered with a single chart.
Example: If <math>M \to N <math> is a covering between manifolds, and <math>F<math> is a foliation on <math>N<math>, then it pulls back to a foliation on <math>M<math>. 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 <math>G<math> is a Lie group, and <math>H<math> is a subgroup obtained by exponentiating a closed subalgebra of the Lie algebra of <math>G<math>, then <math>G<math> is foliated by cosets of <math>H<math>.
There is a close relationship, assuming everything is smooth, with vector fields: given a vector field <math>X<math> on <math>M<math> that is never zero, its integral curves will give a 1-dimensional foliation. (i.e. a codimension <math>n-1<math> 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 <math>n-p<math> 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 <math>GL(n)<math> 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.