# The Unapologetic Mathematician

## Vector Fields

At last, we get back to the differential geometry and topology. Let’s say that we have a manifold $M$ with tangent bundle $\mathcal{T}M$, which of course comes with a projection map $\pi:\mathcal{T}M\to M$. If $U\subseteq M$ is an open submanifold, we can restrict the bundle to the tangent bundle $\pi:\mathcal{T}U\to U$ with no real difficulty.

Now a “vector field” on $U$ is a “section” of this projection map. That is, it’s a function $v:U\to\mathcal{T}U$ so that the composition $\pi\circ v:U\to U$ is the identity map on $U$. In other words, to every point $p\in U$ we get a vector $v(p)\in\mathcal{T}_pU$ 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 $\mathbb{R}^3$ is a function which assigns “a vector” to every point in some region $U\subseteq\mathbb{R}^3$; that is, a function $U\to\mathbb{R}^3$. 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 $M$ has dimension $n$ then each tangent space $\mathcal{T}_pM$ has dimension $n$, and thus they’re all isomorphic. Worse, when working over Euclidean space there is a canonical identification between a tangent space $\mathcal{T}_pE$ and the space $E$ 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 $U\subseteq M$ we have a collection of vector fields, which we will write $\mathfrak{X}_MU$, or $\mathfrak{X}U$ for short. It should be apparent that if $V\subseteq U$ is an open subspace we can restrict a vector field on $U$ to one on $V$, 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 $U$, we can define the sum and scalar multiple of vector fields on $U$ just by defining them pointwise. That is, if $v_1$ and $v_2$ are vector fields on $U$ and $a_1$ and $a_2$ are real scalars, then we define

$\displaystyle\left[a_1v_1+a_2v_2\right](p)=a_1v_1(p)+a_2v_2(p)$

using the addition and scalar multiplication in $\mathcal{T}_pM$. But that’s not all; we can also multiply a vector field $v\in\mathfrak{X}U$ by any function $f\in\mathcal{O}U$:

$\displaystyle\left[fv\right](p)=f(p)v(p)$

using the scalar multiplication in $\mathcal{T}_pM$. This makes $\mathfrak{X}_M$ into a sheaf of modules over the sheaf of rings $\mathcal{O}_M$.