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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.

For $n\ne 0$, put $\rho =f(r)/|n|$ and $\theta =|n|\varphi$. Then $d{\rho }^2=d{r}^2$ andThese give a local isometry wherever $\rho >0$. For $n=0$, $f=b>0$, and the local coordinates $X=r$, $Y=b\varphi$ similarly give the Euclidean plane metric tensor. Equivalently, the Gaussian curvature of the original Riemannian metric is $-f^{\prime \prime }/f=0$ throughout this family.

A local isometry need not be a global isometry. If $|n|=1$, the displayed polar map is injective modulo $\varphi \sim \varphi +2\pi$ and identifies the regular domain with a Euclidean annulus, possibly a punctured disc or the punctured Euclidean plane, according to the radial interval. If $|n|>1$, it is a $|n|$-fold covering of that annulus: distinct angles differing by $2\pi /|n|$ have the same image. This developing map cannot be made injective by a different choice of Euclidean coordinates, since two local Euclidean isometries on a connected flat domain differ by a rigid motion once their value and differential are fixed at one point. For $n=0$, the domain is a flat cylinder; its circular curves are closed geodesics, whereas the Euclidean plane has no nonconstant closed geodesic.

All these metrics are locally Euclidean; they are not all globally Euclidean. Even the regular $|n|=1$ domain is an annular region rather than the entire Euclidean plane. A regular axis can be added without a conical singularity only when $|n|=1$; with the full radial range this completion becomes the Euclidean plane. Trivial holonomy alone therefore does not supply a global injective Euclidean chart.

Writing the upper cone as $(r\cos \theta ,r\sin \theta ,r)$ gives metricOn any polar branch, the mappulls this back to $d{s}^2+s^{2}d{\phi }^2$, so it is a local isometry. Strictly, because $\sqrt{2}\phi$ is not $2\pi$-periodic, this formula is globally single-valued on the universal polar cover and locally on ${\mathbb{R}}^2\setminus \{0\}$; that is the developing map used here.

An isometry fixing $p$ lifts locally to a Euclidean isometry fixing a lift of $p$. Compatibility with the cone's angular identification leaves only the identity and reflection across a radial line. Descending to the cone, these are the identity and reflection in the plane through the cone axis and $p$.