2000, VOLUME 6, NUMBER 1, PAGES 299-303

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

N. E. Maryukova


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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.

All articles are published in Russian.

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Last modified: April 11, 2000