Foliation

In mathematics, a foliation structure on a manifold M gives it stripes. For example if the dimension of M is two, there is a pattern of stripes on the Euclidean plane formed by all lines parallel to the x-axis (lines y = c), and a foliation on M is a consistent way of identifying patches on M with such a striped pattern.

More generally if M has dimension n, a codimension k foliation of M would be a pattern based on the model given in n-dimensional by equations x_i = c_i for i = n-k, n-k_1, ... , n. This is interesting in all cases with 0 < k < n. There is a close relation, assuming everything is smooth, with vector fields: given a vector field X on M that is never zero, its integral curves will give a codimension n-1 foliation.

This observation generalises to a theorem of Frobenius, saying that the necessary conditions, in terms of the Lie bracket, for local co-ordinates for a number of vector fields X_i to have solutions like the n-space model of the correct codimension, are also sufficient. One can also phrase this differently, as a question of reduction of the structure group of the tangent bundle from Gl_n to the subgroup H sending the subspace x_i = 0 (i = 1 to n - k - 1) to itself. 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 require 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 for the torus. That's a consequence of the Hopf index theorem, which shows the Euler characteristic will have to be 0.