Integral curves of vector fields on closed surfaces

Question

If we have a vector field on a boundary less and compact 2-manifold, which is neither a gradient nor a harmonic, does that imply its integral curves are closed?

Answer 1

Let $T$ be the $2$-torus with the flat metric. Then every vector field of the form$$X=a\frac{\partial}{\partial x}+b\frac{\partial}{\partial y},$$where $a$ and $b$ are constants, is harmonic. By the Hodge decomposition theorem, the space of harmonic vector fields is $2$-dimensional, and hence all the harmonic vector fields have the above form.

Let $f:T\to\mathbb{R}$ be smooth, non-vanishing and non-constant. Define a vector field $X$ by$$X(p)=f(p)\frac{\partial}{\partial x}+cf(p)\frac{\partial}{\partial y},$$where $c$ is an irrational constant. Then the trajectories of $X$ are straight lines with irrational slope, hence not closed. Clearly, $X$ is not a gradient, as it does not vanish. By the first paragraph, $X$ is not harmonic.