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.