Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 11F / c / Solution

The helicoid has no angular identification: its height coordinate is $v$. ConsequentlyThe catenoid has total curvature $-2\pi$. A global Riemannian isometry preserves both the Gaussian curvature and the area element, hence preserves total curvature. ThereforeThe same obstruction is visible in the parameter map: increasing $v$ by $2\pi$ gives a different point of the helicoid but the same point of the catenoid.