= Holonomy around circular fibres
{title2=$\mathcal P=\operatorname{Rot}(-2\pi f'(r_0))$}
For the <Riemannian metric> $dr^2+f(r)^2d\phi^2$, where $f>0$ and $\phi$ has period $2\pi$, <parallel transport> around $r=r_0$ acts on the <orthonormal basis> components of a <tangent vector> by a rotation through $-2\pi f'(r_0)$. Indeed, those components obey $u'=f'(r_0)v$ and $v'=-f'(r_0)u$ as functions of $\phi$. Thus all <tangent vectors> return to themselves exactly when $f'(r_0)\in\mathbb Z$. The integer criterion concerns <holonomy> and is stronger than local flatness.
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