# Local isometry that sends principal curves of one surface to asymptotic curves of another surface

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.

• 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).
• You're welcome! Incidentally, a slightly less magical argument for the helicoid would be to check that the helices ($v$-coordinate curves) have acceleration tangent to the surface. – Andrew D. Hwang Dec 29 '16 at 18:16
• The fact that the respective parametrizations have the same first fundamental form means the map $i(X(u, v)) = Y(u, v)$ is a local isometry, as you note. If $\gamma(t) = (u(t), v(t))$ parametrizes a curve in $X$ (e.g., a line of curvature), the image $i(\gamma(t)) = Y(u(t), v(t))$ parametrizes the image. Here you can check explicitly that $t \mapsto Y(t, v_{0})$ and $t \mapsto Y(u_{0}, t)$ are asymptotic curves. – Andrew D. Hwang Dec 30 '16 at 12:50