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On a connected regular radial interval, trivial holonomy around circular fibres forces the continuous function $f^{\prime }$ 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 $-f^{\prime \prime }/f=0$. For $n\ne 0$, $\rho =f/|n|$ and $\theta =|n|\varphi$ give a local isometry to the Euclidean plane; for $n=0$, $r,b\varphi$ 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.

Work on a connected regular radial interval, where $f$ is smooth and nonzero; replacing $f$ by $|f|$ allows us to take $f>0$. The only nonzero Christoffel symbols areAlong the circle, write a tangent vector in the single-valued orthonormal basis $E_r={\partial }_r$, $E_{\varphi }=f^{-1}{\partial }_{\varphi }$ as $V=u{E}_r+v{E}_{\varphi }$. The equations for parallel transport reduce toConsequently $u+iv$ acquires the factor $e^{-2\pi i{f}^{\prime }(r_0)}$ after one circuit. The basis itself agrees at $\varphi =0$ and $2\pi$, so no extra basis change is missing. The holonomy is the identity on every tangent vector precisely whenThe first condition makes this a regular circle of the Riemannian metric; the second is the actual transport condition. In particular, zero slope is sufficient but not necessary. This is holonomy around circular fibres.