Parallel of a surface of revolution

A parallel of a surface of revolution is the circle obtained by fixing the profile parameter and varying the rotation angle. Its radius is the distance $f(v)$ from the rotation axis.

For the arc-length meridian metric $d{s}^2+f(s{)}^{2}d{\theta }^2$, a unit-speed parallel has $\dot{s}=0$ and $\dot{\theta }=\pm 1/f(s_0)$. Its radial geodesic equation is satisfied exactly when $f\prime (s_0)=0$. Thus extrema of the radius, and any other stationary radii, give geodesic circles. If every parallel is geodesic, the surface is a circular cylinder.