FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

Abstract

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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

$$y_{tt} - y_{uu} = M^{2}y
(M = const, y = y(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.

Location: http://mech.math.msu.su/~fpm/eng/k00/k001/k00126h.htm

Last modified: April 11, 2000