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 principal curvature.

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

$$ d \phi_1^2 + d \phi_2^2 = - K ds^2 $$

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