At last, we get back to the differential geometry and topology. Let’s say that we have a manifold with tangent bundle , which of course comes with a projection map . If is an open submanifold, we can restrict the bundle to the tangent bundle with no real difficulty.
Now a “vector field” on is a “section” of this projection map. That is, it’s a function so that the composition is the identity map on . In other words, to every point we get a vector at that point.
I should step aside to dissuade people from a common mistake. Back in multivariable calculus, it’s common to say that a vector field in is a function which assigns “a vector” to every point in some region ; that is, a function . The problem here is that it’s assuming that every point gets a vector in the same vector space, when actually each point gets assigned a vector in its own tangent space.
The confusion comes because we know that if has dimension then each tangent space has dimension , and thus they’re all isomorphic. Worse, when working over Euclidean space there is a canonical identification between a tangent space and the space itself, and thus between any two tangent spaces. But when we’re dealing with an arbitrary manifold there is no such canonical way to compare vectors based at different points; we have to be careful to keep them separate.
For each we have a collection of vector fields, which we will write , or for short. It should be apparent that if is an open subspace we can restrict a vector field on to one on , which means we’re talking about a presheaf. In fact, it’s not hard to see that we can uniquely glue together vector fields which agree on shared domains, meaning we have a sheaf of vector fields.
For any , we can define the sum and scalar multiple of vector fields on just by defining them pointwise. That is, if and are vector fields on and and are real scalars, then we define
using the addition and scalar multiplication in . But that’s not all; we can also multiply a vector field by any function :
using the scalar multiplication in . This makes into a sheaf of modules over the sheaf of rings .