Constant curvature surfaces in the constant curvature quasi-Riemann space and the Klein–Gordon equation

A three-dimensional quasi-Riemann space of constant curvature can be Galilean, quasi-elliptic or quasi-hyperbolic depending on the sign of the curvature. The results obtained by the author for the Galilean case are generalized to the case of quasi-elliptic and quasi-hyperbolic spaces. It is shown that the curvature radius of special lines as well as the angle between asymptotic lines on the surface of constant negative (positive) curvature in quasi-elliptic (quasi-hyperbolic) space satisfy one-dimensional Klein–Gordon equation
$$ \psi_{tt}-\psi_{uu}=M^2\psi\quad (M=\mathrm{const},\ \psi=\psi(t,u)), $$
and, in addition, for the surfaces of quasi-elliptic space, which have Gaussian curvature with absolute value equal to that of the space curvature, $M=0$ in the Klein–Gordon equation. The existence of surfaces corresponding to any given solution of Klein–Gordon equation is shown, the families of surfaces for some special class of such solutions are constructed.