 Holonomy around circular fibres

For the Riemannian metric $d{r}^2+f(r{)}^{2}d{\varphi }^2$, where $f>0$ and $\varphi$ has period $2\pi$, parallel transport around $r=r_0$ acts on the orthonormal basis components of a tangent vector by a rotation through $-2\pi f^{\prime }(r_0)$. Indeed, those components obey $u^{\prime }=f^{\prime }(r_0)v$ and $v^{\prime }=-f^{\prime }(r_0)u$ as functions of $\varphi$. Thus all tangent vectors return to themselves exactly when $f^{\prime }(r_0)\in \mathbb{Z}$. The integer criterion concerns holonomy and is stronger than local flatness.