A question about connections on Riemann Surfaces

Question

I have been self studying calculus on line bundles on Riemann Surfaces from “Riemann Surfaces by Way of Complex Analytic Geometry” Chapter 6 by Dror Varolin. I am trying to understand why the connection form is not globally defined. In the book, it says it follows from the equation for two connection forms $\omega$ and $\omega’$ coming from two different frames on an open set, we have $$ \omega=\omega’+\frac{df}{f} $$ I don’t understand why this implies $w$ cannot be defined globally. We already know $f$ cannot have a zero for it is the “ratio” of two frames. I am somewhat new in the topic so this might be very easy to see. Thanks in advance!

The part in the book