A geometrical interpretation of the sine-Gordon equation (Q1603063)

The author proves the following theorem. In a three-dimensional Riemannian space of constant curvature, let a surface \(V\) of constant negative Gaussian curvature \(K\) be given related to Chebyshev coordinates. If the Gaussian curvature of \(V\) is not equal by the absolute value to the curvature of the space, then the angle \(\psi\) between the asymptotic lines on \(V\) satisfies the sine-Gordon equation \(\psi_{xy} = -K_0 \sin\psi\), where \(K_0\) is the intrinsic curvature of \(V\). If the Gaussian curvature of \(V\) coincides by the absolute value with the curvature of the space, then the angle between the asymptotic lines satisfies the equation \(\psi_{xy} = 0\).