From Encyclopedia of Mathematics
Jump to: navigation, search

A surface of constant negative curvature formed by rotating a tractrix (, ) around its asymptote (; see Fig.).

Figure: p075840a

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

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

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).


[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)
How to Cite This Entry:
Pseudo-sphere. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098