A Chebychev net obeying Sine-Gordon equation is drawn on a surface of constant negative Gauss Curvature $K$ so that the asymptotic differential rhombic element corners lie on lines of maximum/minimum curvature.

Show that principal rotation of surface normals across diagonals ($\phi_1$ = const, $\phi_2 $ = const.) of rhombus are related as:

$$ d \phi_1^2 + d \phi_2^2 = - K ds^2 \tag{0} $$

which is a hyperbolic metric with rotation parameters $\phi_1,\phi_2$

EDIT1:

The following fully reinforces original view of hyperbolic geometry where the parameters $ (\phi_1, \phi_2 )$ take the place of $(u,v)$ in the Euclidean metric upto an invariant constant $a$ :

$$ ds^2= du^2 + dv^2 \tag{1} $$

Let $ K= -1/a^2$ , primes with respect to hyperbolic geodesic asymptotic arcs

$$ \phi_1^{\prime2} +\phi_2^{\prime2} = \frac{1}{a^2} \tag{2}$$

Taking components of arc along inclined direction $\psi$ of fiber to maximum and minimum curvature directions, we have by definition of constant $K$

$$ \frac{d \phi_1}{ds \cos \psi} \cdot \frac{d \phi_2}{ds \sin \psi} = \frac{ \phi_1 ^{\prime}}{ \cos \psi} \cdot \frac{ \phi_2 ^{\prime}}{ \sin \psi} =\frac{1}{a^2} \tag{3} $$

Solving (2),(3) we obtain derivatives of rotation in each direction from as:

$$\phi_1^{\prime} = \frac { \cos \psi} {a}, \;\phi_2^{\prime} = \frac { \sin \psi} {a} \tag{4} $$

as one solution taken out of two interchangeable solutions. Squaring and adding,

$$ ds^2 = a^2( d \phi_1^2 + d \phi_2^2). \tag{5} $$

What prompts me to post this is: Recognition of this observed identity between Euclidean and Hyperbolic parameters to hopefully remove vagueness (in my mind at least) while recognizing these rotations as hyperbolic parameters:

$$ (u,v) \leftrightarrow a (\phi_1 , \phi_2 ) \tag{6} $$

In this connection I quote from the text book of DJ Struik, Lectures on Classical Differential Geometry, Second edition pp 153 left bottom:

The whole geometry of Lobachevski-Bolyai could thus be interpreted on a surface of constant negative curvature , parallel lines becoming geodesics (Emphasis mine). Beltrami proved that the consistency of implied consistency of Lobachevski-Bolyai geometry, since an inconsistency in the latter could be interpreted as an inconsistency in the theory of surfaces of constant negative (Gauss) curvature which itself is based on Euclidean postulates.

Above image is made on Mathematica based on the metric correspondence (6). The discussion is for any surface, not necessarily that of revolution as pictured.

EDIT2:

Derivation:

$ \kappa_{1,2}$ principal curvatures. Euler's normal curvature relation:

$$ \kappa_n =\kappa_1 \cos^2 \psi + \kappa_2 \sin ^2 \psi =0 ;\, \kappa_1 \kappa_2 = -1/a^2 \, \rightarrow \kappa_{1,2}= (-\tan\psi/a, \cot\psi/a) \tag{7}$$

Line segment components along curvature extrema directions :

$$ d \phi_1 = 2 \, ds\, \cos \psi \, \kappa_1,\, d\phi_2 = 2 \, ds \, \sin \psi \, \kappa_2,\ \tag{8} $$

Combining (7),(8) to eliminate $\kappa_{1,2}$ we get (2) or (0).