This is a question on a previous comprehensive exam that I am currently using to study. I can answer the first part, but am not quite sure what to do with the second.
Let $a > 0$ and let $M$ be the catenoid given by the parametrization $$X(u,v) = \left(\sqrt{u^2 + a^2} \cos v, \sqrt{u^2 + a^2} \sin v, a \ln \left(u + \sqrt{u^2 + a^2}\right) \right) $$ and let $N$ be the helicoid given by the parametrization $$Y(u,v) = (u\cos v, u \sin v, av).$$ Show that the map $X(u,v) \mapsto Y(u,v)$ is a local isometry that takes principal curves in $M$ to principal curves in $N$.
I have shown that the first fundamental forms are the same, so the map $X(u,v) \mapsto Y(u,v)$ is in fact a local isometry. However, I am not sure how to show that it takes principal curves in $M$ to asymptotic curves in $N$.
I assume that I should start with a principal curve on $M$ and push it forward using the $\Phi$ given by $\Phi(X(u,v)) = Y(u,v)$, but am not quite sure how to do that.
In this particular example, the coordinate curves of the catenoid clearly map to the coordinate curves of the helocoid. Morover:
The coordinate curves on the catenoid are principal because the catenoid is a surface of rotation (whose principal curves are latitudes and meridians);
The coordinate curves are asymptotic for the helicoid because the $u$-coordinate curves are rulings (acceleration tangent to the surface), the $v$-coordinate curves are orthogonal (image of mutually-orthogonal curves under a local isometry), and the helicoid is a non-planar minimal surface (so it has two orthogonal asymptotic directions at each point).
For all these properties, see for example Elementary Differential Geometry, second revised edition, by Barrett O'Neill, pp. 242–244. (By freakish coincidence, I opened right to page 244.)