Geodesic parallels of a surface of revolution
= Geodesic parallels of a surface of revolution
{title2=$s=s_0\text{ is geodesic}\Longleftrightarrow f\prime(s_0)=0$}
For the arc-length meridian metric $ds^2+f(s)^2d\theta^2$, a unit-speed parallel has $\dot s=0$ and $\dot\theta=\pm1/f(s_0)$. 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.