The Sine-Gordon equation describes varying angles, conserving differential lengths in a mapping with constant Gauss curvature by means of an ODE.
In which conformal mapping (conserving angles), can we vary lengths with constant Gauss curvature?
EDIT1/2:
(It frustrated me much to find no answer in standard DG texts that I could access. Loxodromes on spheres and pseudospheres I consider to be too trivial.)
EDIT 3 :
A bit of research I have so far done:
If $k_1,k_2,k_n,\tau_g$ are respectively principal & normal curvatures, geodesic torsion and $ k_{hs}, k_{hd} $ are half sum (average) and half difference principal curvatures, then
$$ (k_n - k_{hs})^2 + \tau_g ^2 = k_{hd}^2. $$
Attempts to visualize these curves on spheres/ pseudospheres will be highly appreciated I am sure, by the community as well.