A Chebyshev 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 normal 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  $\phi_1,\phi_2$ as parameters.

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 up to 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 direction inclined at $\psi$ to fiber along 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 w.r.t. arc in each direction 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,

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

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:

Thus the above is the ***curvilinear hyperbolic geodesic Pythagoras theorem***. Hypotenuse is allowed only along hyperbolic geodesics and components only along maximum/minimum normal curvature lines.

I have no access to good literature references/ sources but had held this view within myself that... this was Beltrami's original conceptualization. Request enlightened members to please make corrections and give comments on my view. 

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

[![Hyp_Geods_Pseudosphere/metric][1]][1]

In this connection I quote from the text book authored by 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 :

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

Combining (7),(8) to eliminate $\kappa_{1,2}$ we get (5) or (0).

EDIT3:

Constant negative hyperbolic lines $K$ *in the large*  for orthogonal lines *should* accordingly be given by:

$$ \cosh (ds/a) = \cosh d \phi_1\,  \cosh d \phi_2\ $$

and *in the small* by

$$ 1 + (ds/a)^2/2 \approx (1 + d\phi_1^2/2)\,(1 + d\phi_2^2/2) $$

so that we have same:

$$  ds^2 = a^2(d\phi_1^2 + d\phi_2^2 ). \tag{10=5} $$

EDIT 4:

Further on I bring out a simple but an equally important ***isometric identity of rotation product*** $ ( d \phi_1.d \phi_2) $  and second differential of $\psi $ which appears to be implied or hidden.

From (4)

$$ d \phi_1 =\frac {ds \cos \psi}{R_1}= \kappa_1  {ds \cos \psi};\quad d \phi_2 =\frac {ds \sin \psi}{R_2}= \kappa_2  {ds \sin\psi};\quad \tag {11 }$$

$$ d \phi_1 . d \phi_2 =\kappa_1 \kappa_2 \, ds \cos \psi  \,ds \sin \psi  = K ds^2 \sin \psi\, \cos \psi \tag{12} $$ 

By Sine-Gordon theorem for half the scissor angle $ 2 \psi$

$$ \psi^{''}=  \frac{d^2\psi}{ds^2}=K \sin\psi \cos\psi =\kappa_1 \kappa_2 \sin\psi \cos\psi  \tag {13} $$

From (13) and (12) the scalar differential rotation product is:

$$ \boxed{ d^2\psi  = d \phi_1 . d \phi_2} \tag{14} $$


  [1]: https://i.sstatic.net/e7aZe.png