On a double covering there is a differential form $\omega$ arises by the pullback of a differential form under the projection iff it is the pullback of $\omega$ under the map $i$, where $i$ is the map induced from the involution of orientation, $\omega$ itself. Why is this the case?
I ask this question because I am really stuck reading this document.
The step is on page 148 last paragraph. Thanks in advance.