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)
[a1]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2]	H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. 320, 378
[a3]	H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1957)
[a4]	M.J. Greenberg, "Euclidean and non-Euclidean geometries" , Freeman (1974)
[a5]	B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)