# 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) |

#### Comments

#### References

[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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Pseudo-sphere&oldid=48353