Suppose $G$ is a compact connected Lie group and $P$ is a $G$-bundle over $S^1$, $A$ is a connection. Then we can choose a frame such that $A = a d\theta$ where $a\in \mathfrak{g}$ is constant. My question is: can we choose the frame such that $a$ falls in a bounded set in $\mathfrak{g}$ for any connection $A$? 



connections on principal bundles over $S^1$
Yes. Let $g\in G$ be the holonomy of the connection $A$; since $G$ is compact and connected, one can choose a frame (i.e., trivialization of $P$) such that $A = a\,\mathrm{d}\theta$ where $g = \exp(2\pi a)$. Since $G$ is assumed compact, there is a compact set $K\subset\frak{g}$ such that $\exp(2\pi K) = G$. Choose $a$ to lie in $K$ and satisfy $g = \exp(2\pi a)$; this does the job.


