= Flat circular metrics with trivial holonomy
{title2=$f(r)=nr+b,\quad n\in\mathbb Z$}
On a connected regular radial interval, trivial <holonomy around circular fibres> forces the continuous function $f'$ to be integer-valued, hence constant. The resulting <Riemannian metrics> have $f(r)=nr+b$, with $n\in\mathbb Z$ and $f>0$ on the interval. Their <Gaussian curvature> is $-f''/f=0$. For $n\ne0$, $\rho=f/|n|$ and $\theta=|n|\phi$ give a <local isometry> to the <Euclidean plane>; for $n=0$, $r,b\phi$ give a flat cylindrical chart. The polar <developing map> is injective on the regular domain when $|n|=1$, and is a multiple covering when $|n|>1$. Trivial <holonomy> therefore does not imply a global Euclidean chart. The slope is a discrete parameter, not an unrestricted real one.
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