# Pseudo-sphere

A surface of constant negative curvature formed by rotating a tractrix ( $x = u - \mathop{\rm tanh} u$, $y = \mathop{\rm sech} u$) around its asymptote ( $y= 0$; see Fig.). Figure: p075840a

The line element in semi-geodesic coordinates has the form:

$$d s ^ {2} = d u ^ {2} + \cosh ^ {2} \frac{u}{a} \ d v ^ {2} ,\ a = \textrm{ const }$$

(the line $u = 0$ is a geodesic); while in isothermal coordinates it has the form:

$$d s ^ {2} = a ^ {2} \frac{d x ^ {2} + d y ^ {2} }{y ^ {2} } ,\ \ a = \textrm{ const } .$$

Every surface of constant negative curvature can be locally imbedded in the pseudo-sphere. The intrinsic geometry of a pseudo-sphere coincides locally with hyperbolic geometry (see Beltrami interpretation).

How to Cite This Entry:
Pseudo-sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-sphere&oldid=48353
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article