Another Example Sheaf: Vector Fields on Manifolds

There's another classic example of sheaves; this one is restricted to manifolds, rather than general topological spaces. But it provides the key to why we can do calculus on a manifold. For any manifold, there is a sheaf of vector fields over the manifold.

Let's start by explaining what a vector field is. A vector field over a topological space T
is a mapping which associates each point in T with a vector in some euclidean
space ℜⁿ. What the vector field is useful for is defining the concept
of differentiation in the manifold. In normal euclidean space, what a rate of change means is
obvious; for example, in ℜ², a positive derivative at a point means that the curve is increasing - the value of the y coordinate is increasing as the x coordinate increases. This is directly related to the fact that if you want to draw a line from one point a to another point b, you can get a vector pointing from a to b that describes the direction you need to go by subtracting a from b. But in an arbitrary manifold, that basic intuition can fail. For example, on the surface of a sphere, if you place a coordinate system centered on a particular point, an increasing line like y=x will "wrap around" and wind up coming back up to the origin; and a vector based on coordinates will not point in the right direction! A vector field provides a way of describing the rate of change and the way that it varies on the surface of an arbitrary manifold - and that also gives you the ability to determine the direction you need to traverse to get from one point to another. What we're to do is use vector fields to define sections of a sheaf over a manifold.

Suppose we've got a manifold, M, which is smooth and connected. A vector field on M is a function from points in M to vectors in the tangent space to M. The tangent space at a point p in M is a vector space which contains a set of vectors which specify the directions through which a curve can pass through p. Since the vectors are all in the tangent space, and the manifold is smooth, we know that the vectors in the fields will vary smoothly as we move around M.

To define the sheaf, we first need to define the presheaf - that is, the mappings that will define the sheaf. Then we'll show that the presheaf satisfies the normalization and gluing axioms.

So, we've got our smooth manifold, M. The presheaf F(M) needs to map every open set in M to a
set of vector fields. In fact, what we'll do is, for each open set O in M, we'll set
F(O) to the set of all vector fields on O. So for O, F will effectively provide a mapping to a set of functions from points in O to vectors in the tangent space for O. For two open sets O and P in M, where O ⊆ P, the restriction maps ρ_O,P for F will restrict the set of vectors
in O to be the set of vectors from points in O that will pass through points in P.

Now, on to the normalization axiom. F(∅) is the empty function, because ∅ has no points. So it maps to an empty vector field. That works perfectly; the restriction maps will work perfectly, because restricting to F(∅) will do the right thing - there are no paths from points in any open set to points in ∅, so the restriction map will always restrict to ∅ - which is exactly right.

For the gluing axiom, we'll play basically the same trick we did last time - we'll just show that the definition we used to create the space necessarily restricts things so that gluing works. So, let's take a set of open sets in M: U = ∪ { u₁, ..., u_n }. For each u_i, we'll choose our vector fields, so that the set of vector fields on members of U agree on overlaps. Then we define the field on the union, U, so that for each point p in U, we
pick a u_i that contains p. For the field on U, we use the mapping F(u_i) to
provide the mapping for F(U). Since we already restricted the fields so that they agree on overlaps,
this process will give us a valid field with smooth variations - and we're guaranteed to end up
with only one mapping defining the overlap sections, because we defined it to select one of the mappings as the canonical one.

So what does this sheaf tell us? Basically, it tells us that the manifold is differentiable. Why? Because we constructed the sheaf that way. Put that way, it looks circular. The catch is that we relied on certain properties of the topology (that it was a manifold, that it was smooth, that it was connected) to create the sheaf. This would not be a sheaf on a topological space without those properties - so what we've done is shown what properties of a manifold make it differentiable.