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
```

$$

\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.

Location: http://mech.math.msu.su/~fpm/eng/k00/k001/k00126t.htm

Last modified: April 11, 2000