What ways are there to comb a torus?

Question

Modulo diffeomorphisms, what are all possible nonvanishing vector fields on the two-torus?

Answer 1

This approaches an answer, but doesn't completely settle the question.

Classifying nonvanishing vector fields on a torus is at least as complicated as classifying line fields (pass to the subbundle of $T(T^2)$ that the vector field spans) up to diffeomorphism. I don't know whether these problems are equivalent. I have the vague feeling that the answer is yes but no real reason to think so.

Because line fields are one-dimensional, they're integrable. That is, a line field on the torus is the same thing as a codimension 1 foliation.

In particular, there are uncountably many different foliations up to diffeomorphism; Kronecker foliations of the torus with irrational angle $\theta$ are rarely diffeomorphic. See here. In particular, each $T_\theta$ can only be diffeomorphic to countably many others.

By theorem 4.2 here we have a decent idea as to what line fields on tori look like. Let's think about case 2); case 1) is fairly straightforward.

There is a Reeb component. We can pick a diffeomorphism so that this Reeb component is some standard Reeb component (say, the Reeb component is $[0,1/2] \times [0,1]$, thinking of the torus as a quotient of the unit square $[0,1] \times [0,1]$. Now we have to think about the foliations on the annulus $[1/2,1] \times [0,1]/\sim$; that same link says they're unions of Reeb components and suspensions of foliations. This is probably as good a classification as you're going to get.

(One should be a little worried about smoothness here when we're doing diffeomorphisms near the boundaries of each of these annuli. I don't know whether there's a way to make sure these diffeomorphisms are actually smooth everywhere.)