A vector field defines a one-dimensional subspace of at any point with : the subspace spanned by . If is everywhere nonzero, then it defines a one-dimensional subspace of each tangent space. A distribution generalizes this sort of thing to higher dimensions.
To this end, we define a -dimensional distribution on an -dimensional manifold to be a map , where is a -dimensional subspace of . Further, we require that this map be “smooth”, in the sense that for any there exists some neighborhood of and vector fields such that the vectors span for each .
Notice here that the don’t have to work for the whole manifold . Indeed, we will see that in many cases there are no everywhere-nonzero vector fields on a manifold . But over a small patch we might more easily find vector fields that are linearly independent at each point, and thus define a smooth -dimensional distribution over . Then more general smooth distributions come from patching these sorts of smooth distributions together.
A vector field on “belongs to” a distribution — which we write — if for all . We say that is “integrable” if for all and belonging to .
Every one-dimensional manifold is integrable. To see this, we note that if and belong to then for some constant , at least at those points where . Thus we see that
and so is proportional to , and thus belongs to . To handle points where , we can put the scalar multiplier on the other side.