The Unapologetic Mathematician

Mathematics for the interested outsider

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.

May 23, 2011 Posted by | Differential Topology, Topology | 10 Comments

   

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