Lines of curvature on Enneper's surface

Question

I'm trying to find the lines of curvature in Enneper's surface, using the following parametrization:

$$X(u,v)=\left(u-\frac{u^3}{3}+uv^2, -v-u^2v+\frac{v^3}{3},u^2-v^2\right)$$

The first fundamental is determined by:

$E=(1+u^2+v^2)^2$

$F=0$

$G=(1+u^2+v^2)^2$

And the second fudamental form by:

$e=-2$

$f=0$

$g=2$

I only know the definition of line of curvature and the fact that $\alpha:I\to S$ is a line of curvature iff $(N\circ\alpha)'(t)=\lambda(t)\alpha'(t)^{(*)}$, where $N$ is the Gauss map and $\lambda:I\to\mathbb{R}$ is a differentiable function.

With this characterization, I can verify whether or not a given $\alpha$ is a line of curvature, but I have no idea how to come up with an $\alpha$ which satisfies $(*)$.

Any hints?

Answer 1

In general, to find such lines of curvature you will need to solve the following differential equation

$$(fE-eF)(u')^2+(gE-eG)u'v'+(gF-fG)(v')^2=0.$$

You can find this equation in do Carmo's book, page 161, section 3-3. To deduce it, basically use the condition you mentioned in your question and also an explicit formulation for the differential of the Gauss map, also found in the do Carmo's.

I hope it helps.

Answer 2

I know this is an old question but this observation could be useful.
If we denote the coefficients of the Weingarten's matrix $S=\text{I}^{-1}\text{II}\in\operatorname{M}_2(\mathbb R)$ with $(a_{ij})$, then it holds the following result:
coordinate lines are lines of curvature $\iff a_{21}=a_{12}=0\iff F=f=0.$