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).

References

 [1] M.Ya. Vygodskii, "Differential geometry" , Moscow-Leningrad (1949) (In Russian) [2] V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 2 , Moscow-Leningrad (1949) (In Russian)