FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2000, VOLUME 6, NUMBER 1, PAGES 299-303
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.
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Last modified: April 11, 2000