I believe that Catenoida catenoid supports a parametrization $\sigma : U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ that forms a conjugate system (i.e., $\sigma_{uv} \in\mathrm{span}(\sigma_u, \sigma_v)$) with the additional property that the coordinate curves are geodesics. I tried the following computational approach in building such a prameterization. I chose the following usual parameterization of a catenoid:
$$ \sigma = \left(\begin{array}{c} \cosh(u) \cos(v),\\ \cosh(u) \sin(v),\\ u\end{array}\right) $$
I assumed that there is a coordinate transformation $u = \phi(x,y)$ and $ v = \psi(x,y)$ such that they would give me the result I need. Upon computing the following conditions that ensure the geodesic-conjugate properties
- conjugate property $\det(\sigma_x,\sigma_y,\sigma_{xy}) = 0$
- geodesic x-coordinates $\det(N,\sigma_x,\sigma_{xx}) = 0$
- geodesic y-coordinates $\det(N,\sigma_y,\sigma_{yy}) = 0$
where $N$ is the Gauss map, I arrived to the following system of PDEs:
$\ln \left(\psi_{x}\right)-\ln \left(\phi_{x}\right) = \ln \left(\psi_{y}\right)-\ln \left(\phi_{y}\right) $
$\frac{{\partial}}{{\partial}y}\ln \left(\psi_{y}\right)-\frac{{\partial}}{{\partial}y}\ln \left(\phi_{y}\right) = \frac{\left(\phi_{y}^{2}+\psi_{y}^{2}\right) \tanh \left(\phi \right)}{\phi_{y}}$
$\frac{{\partial}}{{\partial}x}\ln \left(\psi_{x}\right)-\frac{{\partial}}{{\partial}x}\ln \left(\phi_{x}\right)=\frac{\left(\phi_{x}^{2}+\psi_{x}^{2}\right) \tanh\left(\phi \right)}{\phi_{x}}$
Can someone clarify how I can continue from here? Unfortunately I don't know much about PDEs but I still believe that there should be an answer to this.
