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A surface of constant negative curvature formed by rotating a [[Tractrix|tractrix]] (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075840/p0758401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075840/p0758402.png" />) around its asymptote (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075840/p0758403.png" />; see Fig.).
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A surface of constant negative curvature formed by rotating a [[Tractrix|tractrix]] ( $  x = u -  \mathop{\rm tanh}  u $,  
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$  y = \mathop{\rm sech}  u $)  
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around its asymptote ( $  y= 0 $;  
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see Fig.).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p075840a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p075840a.gif" />
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The line element in [[Semi-geodesic coordinates|semi-geodesic coordinates]] has the form:
 
The line element in [[Semi-geodesic coordinates|semi-geodesic coordinates]] has the form:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075840/p0758404.png" /></td> </tr></table>
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$$
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d s  ^ {2}  = d u  ^ {2} + \cosh  ^ {2} 
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\frac{u}{a}
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\
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d v  ^ {2} ,\  a = \textrm{ const }
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$$
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(the line  $  u = 0 $
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is a geodesic); while in [[Isothermal coordinates|isothermal coordinates]] it has the form:
  
(the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075840/p0758405.png" /> is a geodesic); while in [[Isothermal coordinates|isothermal coordinates]] it has the form:
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$$
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d s  ^ {2}  = a ^ {2}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075840/p0758406.png" /></td> </tr></table>
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\frac{d x  ^ {2} + d y  ^ {2} }{y  ^ {2} }
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,\ \
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a = \textrm{ const } .
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$$
  
 
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|Beltrami interpretation]]).
 
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|Beltrami interpretation]]).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Ya. Vygodskii,  "Differential geometry" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.F. Kagan,  "Foundations of the theory of surfaces in a tensor setting" , '''2''' , Moscow-Leningrad  (1949)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Ya. Vygodskii,  "Differential geometry" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.F. Kagan,  "Foundations of the theory of surfaces in a tensor setting" , '''2''' , Moscow-Leningrad  (1949)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1969)  pp. 320, 378</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.S.M. Coxeter,  "Non-Euclidean geometry" , Univ. Toronto Press  (1957)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.J. Greenberg,  "Euclidean and non-Euclidean geometries" , Freeman  (1974)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.A. [B.A. Rozenfel'd] Rosenfel'd,  "A history of non-euclidean geometry" , Springer  (1988)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1969)  pp. 320, 378</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.S.M. Coxeter,  "Non-Euclidean geometry" , Univ. Toronto Press  (1957)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.J. Greenberg,  "Euclidean and non-Euclidean geometries" , Freeman  (1974)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.A. [B.A. Rozenfel'd] Rosenfel'd,  "A history of non-euclidean geometry" , Springer  (1988)  (Translated from Russian)</TD></TR></table>

Revision as of 08:08, 6 June 2020


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
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article