When do parallel sections exist?

Question

I suspect that this is a "trivial" question, but I don't have enough background to know the answer immediately: Suppose $\pi : E \to M$ is a trivial real line bundle on a smooth manifold $M$, and suppose we have a flat connection $\nabla$ on $E$. Does there always exist a nowhere vanishing parallel section of $E$? If not, when does one exist?

Answer 1

Write $\nabla = d + a$, where $a$ is a one form. Then if $\nabla s = 0$ (here $s$ is a section on the trivial bundle, thus is a function $s: M \to \mathbb R$), we have

$$ds + as = 0 \Rightarrow a = -\frac{1}{s} ds = -d (\log s).$$

(Note $s$ is nowhere vanishing, we can assume that it's positive). Thus $a$ is exact.

On the other hand, if $a = df$ is exact, then $s = e^{-f}$ satisfies

$$\nabla s = (d + a) s = 0.$$

Thus $\nabla$ has a nowhere vanishing section if and only if $a$ is exact.

As a result, there are flat connection that has no nontrivial parallel sections. For example, $\nabla = d + d\theta$ on the trivial bundle of $\mathbb S^1$ does not have such a section.

Note that $a$ is exact is a stronger condition then that $\nabla$ is flat ($\Leftrightarrow a$ is closed).