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A space whose geometry is defined by the axioms of [[Lobachevskii geometry|Lobachevskii geometry]]. In a wider sense a Lobachevskii space is a non-Euclidean hyperbolic space whose definition is connected with concepts of the geometry of a [[Pseudo-Euclidean space|pseudo-Euclidean space]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l0600501.png" /> be the Lorentz–Minkowskian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l0600502.png" />-space with one time-like direction. A sphere of time-like radius is analogous to a hyperboloid of two sheets. One sheet (say the  "future"  sheet) is isometric to a Lobachevskii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l0600503.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l0600504.png" />. This definition of a Lobachevskii space makes it possible to include this space in the projective classification of non-Euclidean spaces. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l0600505.png" /> in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l0600506.png" /> is represented by the interior of an oval (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l0600507.png" />)-quadric that is the intersection of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l0600508.png" />-sphere of time-like radius with the hyperplane at infinity of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l0600509.png" /> that completes this space to the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005010.png" />. The points of the oval (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005011.png" />)-quadric are the points at infinity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005012.png" />, that is, the quadric is the [[Absolute|absolute]] of this space. The outside of this quadric, which completes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005013.png" /> to the complete space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005014.png" />, is called the ideal domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005015.png" />. This interpretation is called the Cayley–Klein projective interpretation. It can also be obtained by projecting an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005016.png" />-sphere of time-like radius in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005017.png" /> from its centre to a tangent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005018.png" />-plane, which is a Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005019.png" />-space; the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005020.png" /> is represented by the inside of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005021.png" />-ball in this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005022.png" />-plane, and the boundary of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005023.png" />-ball is the absolute of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005024.png" /> (the latter interpretation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005025.png" /> in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005026.png" /> is sometimes called the Beltrami–Klein interpretation).
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The projective interpretation of Lobachevskii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005027.png" />-space makes it possible to verify the axioms of Lobachevskii geometry, to give a representation of all figures of this geometry, and to establish their properties; in particular, in this interpretation it is easy to establish the geometrical properties of the Lobachevskii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005028.png" />-plane that follow from the axioms of Lobachevskii geometry.
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When the hyperbolic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005029.png" /> is imbedded in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005030.png" />, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005031.png" />-flat <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005032.png" /> is said to be proper if it intersects the absolute in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005034.png" />-quadric; an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005035.png" />-flat that touches the absolute is isotropic; and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005037.png" />-flat that does not intersect the absolute is ideal. The poles of proper hyperplanes are ideal points, and the proper points are the poles of ideal hyperplanes. More generally, the polar (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005039.png" />)-flats of proper <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005040.png" />-flats of the Lobachevskii space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005041.png" /> are ideal (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005042.png" />)-flats, and the polar (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005043.png" />)-flats of ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005044.png" />-flats are proper (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005045.png" />)-flats.
+
A space whose geometry is defined by the axioms of [[Lobachevskii geometry|Lobachevskii geometry]]. In a wider sense a Lobachevskii space is a non-Euclidean hyperbolic space whose definition is connected with concepts of the geometry of a [[Pseudo-Euclidean space|pseudo-Euclidean space]]. Let  $  {}  ^ {1} R _ {n+} 1 $
 +
be the Lorentz–Minkowskian  $  ( n + 1 ) $-
 +
space with one time-like direction. A sphere of time-like radius is analogous to a hyperboloid of two sheets. One sheet (say the  "future" sheet) is isometric to a Lobachevskii  $  n $-
 +
space  $  {}  ^ {1} S _ {n} $.  
 +
This definition of a Lobachevskii space makes it possible to include this space in the projective classification of non-Euclidean spaces. The space  $  {}  ^ {1} S _ {n} $
 +
in the projective space  $  P _ {n} $
 +
is represented by the interior of an oval ( $  n- 1 $)-
 +
quadric that is the intersection of an $  n $-
 +
sphere of time-like radius with the hyperplane at infinity of the space  $  {}  ^ {1} R _ {n+} 1 $
 +
that completes this space to the projective space  $  P _ {n+} 1 $.  
 +
The points of the oval ( $  n- 1 $)-
 +
quadric are the points at infinity of  $  {}  ^ {1} S _ {n} $,
 +
that is, the quadric is the [[Absolute|absolute]] of this space. The outside of this quadric, which completes  $  {}  ^ {1} S _ {n} $
 +
to the complete space  $  P _ {n} $,
 +
is called the ideal domain of  $  {}  ^ {1} S _ {n} $.  
 +
This interpretation is called the Cayley–Klein projective interpretation. It can also be obtained by projecting an  $  n $-
 +
sphere of time-like radius in  $  {}  ^ {1} R _ {n+} 1 $
 +
from its centre to a tangent  $  n $-
 +
plane, which is a Euclidean  $  n $-
 +
space; the space $  {}  ^ {1} S _ {n} $
 +
is represented by the inside of an  $  n $-
 +
ball in this  $  n $-
 +
plane, and the boundary of the  $  n $-
 +
ball is the absolute of $  {}  ^ {1} S _ {n} $(
 +
the latter interpretation of  $  {}  ^ {1} S _ {n} $
 +
in the Euclidean space  $  R _ {n} $
 +
is sometimes called the Beltrami–Klein interpretation).
  
In the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005046.png" />, as coordinates of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005047.png" /> one takes the components of the corresponding vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005048.png" /> of this point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005049.png" />. These [[Weierstrass coordinates|Weierstrass coordinates]] must satisfy the condition
+
The projective interpretation of Lobachevskii  $  3 $-
 +
space makes it possible to verify the axioms of Lobachevskii geometry, to give a representation of all figures of this geometry, and to establish their properties; in particular, in this interpretation it is easy to establish the geometrical properties of the Lobachevskii  $  2 $-
 +
plane that follow from the axioms of Lobachevskii geometry.
  
<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/l/l060/l060050/l06005050.png" /></td> </tr></table>
+
When the hyperbolic space  $  {}  ^ {1} S _ {n} $
 +
is imbedded in the projective space  $  P _ {n} $,
 +
an  $  m $-
 +
flat  $  ( m < n ) $
 +
is said to be proper if it intersects the absolute in an  $  ( m - 1 ) $-
 +
quadric; an  $  m $-
 +
flat that touches the absolute is isotropic; and an  $  m $-
 +
flat that does not intersect the absolute is ideal. The poles of proper hyperplanes are ideal points, and the proper points are the poles of ideal hyperplanes. More generally, the polar ( $  n - m - 1 $)-
 +
flats of proper  $  m $-
 +
flats of the Lobachevskii space  $  {}  ^ {1} S _ {n} $
 +
are ideal ( $  n- m- 1 $)-
 +
flats, and the polar ( $  n- m- 1 $)-
 +
flats of ideal  $  m $-
 +
flats are proper ( $  n- m- 1 $)-
 +
flats.
  
In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005051.png" /> one may introduce instead coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005052.png" />, analogous to spherical polar coordinates, which are connected with the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005053.png" /> by the relations
+
In the space  $  {}  ^ {1} S _ {n} $,
 +
as coordinates of a point  $  X $
 +
one takes the components of the corresponding vector  $  \mathbf x $
 +
of this point in  $  {}  ^ {1} R _ {n+} 1 $.  
 +
These [[Weierstrass coordinates|Weierstrass coordinates]] must satisfy the condition
  
<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/l/l060/l060050/l06005054.png" /></td> </tr></table>
+
$$
 +
( x  ^ {0} )  ^ {2} - \sum _ { t } ( x  ^ {t} )  ^ {2}  = 1 ,\  t > 0 .
 +
$$
  
<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/l/l060/l060050/l06005055.png" /></td> </tr></table>
+
In  $  {}  ^ {1} S _ {n} $
 +
one may introduce instead coordinates  $  u  ^ {1} \dots u  ^ {n} $,
 +
analogous to spherical polar coordinates, which are connected with the coordinates  $  x  ^ {t} $
 +
by the relations
  
<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/l/l060/l060050/l06005056.png" /></td> </tr></table>
+
$$
 +
x  ^ {0= \cosh  u  ^ {1}  \cosh  u  ^ {2} \dots \cosh  u  ^ {n} ,
 +
$$
  
<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/l/l060/l060050/l06005057.png" /></td> </tr></table>
+
$$
 +
x  ^ {1}  = \sinh  u  ^ {1}  \cosh  u  ^ {2} \dots \cosh  u  ^ {n} ,
 +
$$
  
<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/l/l060/l060050/l06005058.png" /></td> </tr></table>
+
$$
 +
{\dots \dots \dots }
 +
$$
  
<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/l/l060/l060050/l06005059.png" /></td> </tr></table>
+
$$
 +
x  ^ {t}  = \sinh  u  ^ {t}  \cosh  u  ^ {t+} 1 \dots \cosh  u  ^ {n} ,
 +
$$
  
The distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005060.png" /> between two points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005061.png" /> is then defined, in terms of their Weierstrass coordinates, by
+
$$
 +
{\dots \dots \dots \dots }
 +
$$
  
<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/l/l060/l060050/l06005062.png" /></td> </tr></table>
+
$$
 +
x  ^ {n}  = \sinh  u  ^ {n} ,
 +
$$
  
The angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005063.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005064.png" />) between two intersecting hyperplanes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005066.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005067.png" />) can be identified with the space-like distance between their poles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005069.png" />, and is thus given by
+
The distance  $  \delta $
 +
between two points of  $  {}  ^ {1} S _ {n} $
 +
is then defined, in terms of their Weierstrass coordinates, by
  
<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/l/l060/l060050/l06005070.png" /></td> </tr></table>
+
$$
 +
\cosh  \delta  = x  ^ {0} y  ^ {0} - \sum x  ^ {t} y  ^ {t} .
 +
$$
  
Similarly, the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005071.png" /> between two ultraparallel hyperplanes is given by
+
The angle  $  \phi $(
 +
$  \leq  \pi / 2 $)
 +
between two intersecting hyperplanes  $  X _ {0} x  ^ {0} + \sum X _ {t} x _ {t} = 0 $
 +
and  $  Y _ {0} x  ^ {0} + \sum Y _ {t} x  ^ {t} = 0 $(
 +
where  $  - X _ {0}  ^ {2} + \sum X _ {t}  ^ {2} = 1 = - Y _ {0}  ^ {2} + \sum Y _ {t}  ^ {2} $)
 +
can be identified with the space-like distance between their poles  $  ( - X _ {0} , X _ {1} \dots X _ {n} ) $
 +
and  $  ( - Y _ {0} , Y _ {1} \dots Y _ {n} ) $,
 +
and is thus given by
  
<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/l/l060/l060050/l06005072.png" /></td> </tr></table>
+
$$
 +
\cos  \phi  = | - X _ {0} Y _ {0} + \sum X _ {t} Y _ {t} | .
 +
$$
 +
 
 +
Similarly, the distance  $  \delta $
 +
between two ultraparallel hyperplanes is given by
 +
 
 +
$$
 +
\cosh  \delta  = | - X _ {0} Y _ {0} + \sum X _ {t} Y _ {t} | .
 +
$$
  
 
The distance between points and the values of the angles between planes admit expressions in terms of the cross ratios (cf. [[Cross ratio|Cross ratio]]) of points, using points of the absolute.
 
The distance between points and the values of the angles between planes admit expressions in terms of the cross ratios (cf. [[Cross ratio|Cross ratio]]) of points, using points of the absolute.
  
In the Lobachevskii space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005073.png" /> one can define spheres (balls), equi-distant surfaces, horospheres (horocycles for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005074.png" />, cf. [[Horocycle|Horocycle]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005075.png" />-simplexes, etc.
+
In the Lobachevskii space $  {}  ^ {1} S _ {n} $
 +
one can define spheres (balls), equi-distant surfaces, horospheres (horocycles for $  n= 2 $,  
 +
cf. [[Horocycle|Horocycle]]), $  m $-
 +
simplexes, etc.
  
The classification of motions of the Lobachevskii space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005076.png" /> as collineations that take points of the absolute (oval quadric) into itself reduces to the classification of motions fixing one point of the Lorentz–Minkowskian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005077.png" /> without interchanging  "future"  and  "past" . (This is a [[Lie group|Lie group]].) In order to specify a motion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005078.png" /> it is sufficient to give the images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005079.png" /> points that do not lie in one hyperplane.
+
The classification of motions of the Lobachevskii space $  {}  ^ {1} S _ {n} $
 +
as collineations that take points of the absolute (oval quadric) into itself reduces to the classification of motions fixing one point of the Lorentz–Minkowskian space $  {}  ^ {1} R _ {n+} 1 $
 +
without interchanging  "future"  and  "past" . (This is a [[Lie group|Lie group]].) In order to specify a motion of $  {}  ^ {1} S _ {n} $
 +
it is sufficient to give the images of $  n+ 1 $
 +
points that do not lie in one hyperplane.
  
There are several conformal interpretations of a Lobachevskii space, one of which is the [[Poincaré model|Poincaré model]]. It is also possible to have a conformal interpretation of the space on one of its hyperplanes. Apart from these there are interpretations in complex spaces. In particular, for the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005080.png" /> one can construct the [[Kotel'nikov interpretation|Kotel'nikov interpretation]] of manifolds of lines.
+
There are several conformal interpretations of a Lobachevskii space, one of which is the [[Poincaré model|Poincaré model]]. It is also possible to have a conformal interpretation of the space on one of its hyperplanes. Apart from these there are interpretations in complex spaces. In particular, for the space $  {}  ^ {1} S _ {n} $
 +
one can construct the [[Kotel'nikov interpretation|Kotel'nikov interpretation]] of manifolds of lines.
  
By means of projective interpretations, quadrics in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005081.png" />, and particularly in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005082.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005083.png" />, can be classified more completely.
+
By means of projective interpretations, quadrics in $  {}  ^ {1} S _ {n} $,  
 +
and particularly in the $  2 $-
 +
plane $  {}  ^ {1} S _ {2} $,  
 +
can be classified more completely.
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005084.png" /> is a Riemannian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005085.png" />-space of constant negative curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005086.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005087.png" /> is the radius of curvature of the space. The geometry of a Lobachevskii space in sufficiently small neighbourhoods of points is close to the geometry of the Euclidean space of the same dimension.
+
The space $  {}  ^ {1} S _ {n} $
 +
is a Riemannian $  n $-
 +
space of constant negative curvature $  - 1 / \sigma  ^ {2} $,  
 +
where $  \sigma i $
 +
is the radius of curvature of the space. The geometry of a Lobachevskii space in sufficiently small neighbourhoods of points is close to the geometry of the Euclidean space of the same dimension.
  
In the large, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005088.png" /> is homeomorphic to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005089.png" />; it extends indefinitely in all directions. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005090.png" />-flat of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005092.png" />, is a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005093.png" />. Also, straight lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005094.png" /> are geodesics, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005095.png" />-flats are totally geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005096.png" />-surfaces of this space.
+
In the large, the space $  {}  ^ {1} S _ {n} $
 +
is homeomorphic to the space $  R _ {n} $;  
 +
it extends indefinitely in all directions. Any $  m $-
 +
flat of $  {}  ^ {1} S _ {n} $,  
 +
$  m < n $,  
 +
is a space $  {}  ^ {1} S _ {m} $.  
 +
Also, straight lines of $  {}  ^ {1} S _ {n} $
 +
are geodesics, and $  m $-
 +
flats are totally geodesic $  m $-
 +
surfaces of this space.
  
In the projective classification of metrics of non-Euclidean spaces a Lobachevskii space is also classified with respect to the metrics on lines, pencils of planes and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005097.png" />-flats. In particular, on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060050/l06005098.png" />-flat of a Lobachevskii space the projective metric on a line is hyperbolic, and the metric in pencils of lines is elliptic.
+
In the projective classification of metrics of non-Euclidean spaces a Lobachevskii space is also classified with respect to the metrics on lines, pencils of planes and $  m $-
 +
flats. In particular, on a $  2 $-
 +
flat of a Lobachevskii space the projective metric on a line is hyperbolic, and the metric in pencils of lines is elliptic.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Efimov,  "Höhere Geometrie" , Deutsch. Verlag Wissenschaft.  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Klein,  "Vorlesungen über Nicht-Euklidische Geometrie" , Springer  (1928)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.F. Kagan,  "Foundations of geometry" , '''1–2''' , Moscow-Leningrad  (1949–1956)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Efimov,  "Höhere Geometrie" , Deutsch. Verlag Wissenschaft.  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Klein,  "Vorlesungen über Nicht-Euklidische Geometrie" , Springer  (1928)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.F. Kagan,  "Foundations of geometry" , '''1–2''' , Moscow-Leningrad  (1949–1956)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.A. [B.A. Rozenfel'd] Rosenfel'd,  "A history of non-euclidean geometry" , Springer  (1988)  pp. Chapt. 6  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.A. Robb,  "Geometry of time and space" , Cambridge Univ. Press  (1936)  pp. 406</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.S.M. Coxeter,  "Non-Euclidean geometry" , Univ. Toronto Press  (1965)  pp. 209</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Busemann,  P.J. Kelly,  "Projective geometry and projective metrics" , Acad. Press  (1953)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.A. [B.A. Rozenfel'd] Rosenfel'd,  "A history of non-euclidean geometry" , Springer  (1988)  pp. Chapt. 6  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.A. Robb,  "Geometry of time and space" , Cambridge Univ. Press  (1936)  pp. 406</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.S.M. Coxeter,  "Non-Euclidean geometry" , Univ. Toronto Press  (1965)  pp. 209</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Busemann,  P.J. Kelly,  "Projective geometry and projective metrics" , Acad. Press  (1953)</TD></TR></table>

Revision as of 22:17, 5 June 2020


A space whose geometry is defined by the axioms of Lobachevskii geometry. In a wider sense a Lobachevskii space is a non-Euclidean hyperbolic space whose definition is connected with concepts of the geometry of a pseudo-Euclidean space. Let $ {} ^ {1} R _ {n+} 1 $ be the Lorentz–Minkowskian $ ( n + 1 ) $- space with one time-like direction. A sphere of time-like radius is analogous to a hyperboloid of two sheets. One sheet (say the "future" sheet) is isometric to a Lobachevskii $ n $- space $ {} ^ {1} S _ {n} $. This definition of a Lobachevskii space makes it possible to include this space in the projective classification of non-Euclidean spaces. The space $ {} ^ {1} S _ {n} $ in the projective space $ P _ {n} $ is represented by the interior of an oval ( $ n- 1 $)- quadric that is the intersection of an $ n $- sphere of time-like radius with the hyperplane at infinity of the space $ {} ^ {1} R _ {n+} 1 $ that completes this space to the projective space $ P _ {n+} 1 $. The points of the oval ( $ n- 1 $)- quadric are the points at infinity of $ {} ^ {1} S _ {n} $, that is, the quadric is the absolute of this space. The outside of this quadric, which completes $ {} ^ {1} S _ {n} $ to the complete space $ P _ {n} $, is called the ideal domain of $ {} ^ {1} S _ {n} $. This interpretation is called the Cayley–Klein projective interpretation. It can also be obtained by projecting an $ n $- sphere of time-like radius in $ {} ^ {1} R _ {n+} 1 $ from its centre to a tangent $ n $- plane, which is a Euclidean $ n $- space; the space $ {} ^ {1} S _ {n} $ is represented by the inside of an $ n $- ball in this $ n $- plane, and the boundary of the $ n $- ball is the absolute of $ {} ^ {1} S _ {n} $( the latter interpretation of $ {} ^ {1} S _ {n} $ in the Euclidean space $ R _ {n} $ is sometimes called the Beltrami–Klein interpretation).

The projective interpretation of Lobachevskii $ 3 $- space makes it possible to verify the axioms of Lobachevskii geometry, to give a representation of all figures of this geometry, and to establish their properties; in particular, in this interpretation it is easy to establish the geometrical properties of the Lobachevskii $ 2 $- plane that follow from the axioms of Lobachevskii geometry.

When the hyperbolic space $ {} ^ {1} S _ {n} $ is imbedded in the projective space $ P _ {n} $, an $ m $- flat $ ( m < n ) $ is said to be proper if it intersects the absolute in an $ ( m - 1 ) $- quadric; an $ m $- flat that touches the absolute is isotropic; and an $ m $- flat that does not intersect the absolute is ideal. The poles of proper hyperplanes are ideal points, and the proper points are the poles of ideal hyperplanes. More generally, the polar ( $ n - m - 1 $)- flats of proper $ m $- flats of the Lobachevskii space $ {} ^ {1} S _ {n} $ are ideal ( $ n- m- 1 $)- flats, and the polar ( $ n- m- 1 $)- flats of ideal $ m $- flats are proper ( $ n- m- 1 $)- flats.

In the space $ {} ^ {1} S _ {n} $, as coordinates of a point $ X $ one takes the components of the corresponding vector $ \mathbf x $ of this point in $ {} ^ {1} R _ {n+} 1 $. These Weierstrass coordinates must satisfy the condition

$$ ( x ^ {0} ) ^ {2} - \sum _ { t } ( x ^ {t} ) ^ {2} = 1 ,\ t > 0 . $$

In $ {} ^ {1} S _ {n} $ one may introduce instead coordinates $ u ^ {1} \dots u ^ {n} $, analogous to spherical polar coordinates, which are connected with the coordinates $ x ^ {t} $ by the relations

$$ x ^ {0} = \cosh u ^ {1} \cosh u ^ {2} \dots \cosh u ^ {n} , $$

$$ x ^ {1} = \sinh u ^ {1} \cosh u ^ {2} \dots \cosh u ^ {n} , $$

$$ {\dots \dots \dots } $$

$$ x ^ {t} = \sinh u ^ {t} \cosh u ^ {t+} 1 \dots \cosh u ^ {n} , $$

$$ {\dots \dots \dots \dots } $$

$$ x ^ {n} = \sinh u ^ {n} , $$

The distance $ \delta $ between two points of $ {} ^ {1} S _ {n} $ is then defined, in terms of their Weierstrass coordinates, by

$$ \cosh \delta = x ^ {0} y ^ {0} - \sum x ^ {t} y ^ {t} . $$

The angle $ \phi $( $ \leq \pi / 2 $) between two intersecting hyperplanes $ X _ {0} x ^ {0} + \sum X _ {t} x _ {t} = 0 $ and $ Y _ {0} x ^ {0} + \sum Y _ {t} x ^ {t} = 0 $( where $ - X _ {0} ^ {2} + \sum X _ {t} ^ {2} = 1 = - Y _ {0} ^ {2} + \sum Y _ {t} ^ {2} $) can be identified with the space-like distance between their poles $ ( - X _ {0} , X _ {1} \dots X _ {n} ) $ and $ ( - Y _ {0} , Y _ {1} \dots Y _ {n} ) $, and is thus given by

$$ \cos \phi = | - X _ {0} Y _ {0} + \sum X _ {t} Y _ {t} | . $$

Similarly, the distance $ \delta $ between two ultraparallel hyperplanes is given by

$$ \cosh \delta = | - X _ {0} Y _ {0} + \sum X _ {t} Y _ {t} | . $$

The distance between points and the values of the angles between planes admit expressions in terms of the cross ratios (cf. Cross ratio) of points, using points of the absolute.

In the Lobachevskii space $ {} ^ {1} S _ {n} $ one can define spheres (balls), equi-distant surfaces, horospheres (horocycles for $ n= 2 $, cf. Horocycle), $ m $- simplexes, etc.

The classification of motions of the Lobachevskii space $ {} ^ {1} S _ {n} $ as collineations that take points of the absolute (oval quadric) into itself reduces to the classification of motions fixing one point of the Lorentz–Minkowskian space $ {} ^ {1} R _ {n+} 1 $ without interchanging "future" and "past" . (This is a Lie group.) In order to specify a motion of $ {} ^ {1} S _ {n} $ it is sufficient to give the images of $ n+ 1 $ points that do not lie in one hyperplane.

There are several conformal interpretations of a Lobachevskii space, one of which is the Poincaré model. It is also possible to have a conformal interpretation of the space on one of its hyperplanes. Apart from these there are interpretations in complex spaces. In particular, for the space $ {} ^ {1} S _ {n} $ one can construct the Kotel'nikov interpretation of manifolds of lines.

By means of projective interpretations, quadrics in $ {} ^ {1} S _ {n} $, and particularly in the $ 2 $- plane $ {} ^ {1} S _ {2} $, can be classified more completely.

The space $ {} ^ {1} S _ {n} $ is a Riemannian $ n $- space of constant negative curvature $ - 1 / \sigma ^ {2} $, where $ \sigma i $ is the radius of curvature of the space. The geometry of a Lobachevskii space in sufficiently small neighbourhoods of points is close to the geometry of the Euclidean space of the same dimension.

In the large, the space $ {} ^ {1} S _ {n} $ is homeomorphic to the space $ R _ {n} $; it extends indefinitely in all directions. Any $ m $- flat of $ {} ^ {1} S _ {n} $, $ m < n $, is a space $ {} ^ {1} S _ {m} $. Also, straight lines of $ {} ^ {1} S _ {n} $ are geodesics, and $ m $- flats are totally geodesic $ m $- surfaces of this space.

In the projective classification of metrics of non-Euclidean spaces a Lobachevskii space is also classified with respect to the metrics on lines, pencils of planes and $ m $- flats. In particular, on a $ 2 $- flat of a Lobachevskii space the projective metric on a line is hyperbolic, and the metric in pencils of lines is elliptic.

References

[1] N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)
[2] F. Klein, "Vorlesungen über Nicht-Euklidische Geometrie" , Springer (1928)
[3] V.F. Kagan, "Foundations of geometry" , 1–2 , Moscow-Leningrad (1949–1956) (In Russian)
[4] B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)

Comments

References

[a1] B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) pp. Chapt. 6 (Translated from Russian)
[a2] A.A. Robb, "Geometry of time and space" , Cambridge Univ. Press (1936) pp. 406
[a3] H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 209
[a4] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a5] H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953)
How to Cite This Entry:
Lobachevskii space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lobachevskii_space&oldid=17548
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article