Yes indeed, once one deals with this case, then one may look at a quotient of $G$ by a discrete subgroup $\Gamma\subseteq G$. Then any frame on $G$ may be pushed down to a frame on $G/\Gamma$

So I thought about the example about the solid cylinder in $R^3$ (of length $1$) along the $x$-axis and center at $(0,0,0)$ where $f$ is the distance between the projection of a point $P$ on $x$ and $(0,0,0)$. But I found this example a bit artificial since in this case, $N$ is really just $[-1/2,1/2]$ since the fibers outside this interval are empty. But now I just realize that one may identify the two end points of that interval to get a circle. Ok, I see so anything can happen. I guess this answers my question.

I see, so hee simply meant algebraic over the base field of the abelian variety. Thanks.