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A projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q0765301.png" />-space in which a metric is defined by an [[Absolute|absolute]], consisting of an absolute cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q0765302.png" /> of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q0765303.png" /> with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q0765304.png" />-vertex (an absolute plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q0765305.png" />) and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q0765306.png" />-quadric (an absolute quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q0765307.png" />) of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q0765308.png" /> on this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q0765309.png" />-plane. Such a space is called a quasi-hyperbolic space of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653011.png" />, and is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653013.png" />. A quasi-hyperbolic space is a particular case of a [[Semi-hyperbolic space|semi-hyperbolic space]]. The quasi-hyperbolic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653014.png" /> is obtained from the hyperbolic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653015.png" /> by passing to the limit in such a way that the absolute of the hyperbolic space is transformed to the absolute of the quasi-hyperbolic space.
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When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653016.png" />, the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653017.png" /> is a pair of coincident planes that are the same as the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653018.png" />, while the absolute of the space is the same as that of the [[Pseudo-Euclidean space|pseudo-Euclidean space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653019.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653020.png" />, the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653021.png" /> is a pair of real planes; in particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653022.png" /> the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653023.png" /> is the line of intersection of these two planes, while the quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653024.png" /> is a pair of points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653025.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653026.png" />, the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653027.png" /> has point vertex and the absolute of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653028.png" /> is the same as that of the [[Co-pseudo-Euclidean space|co-pseudo-Euclidean space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653029.png" />.
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A projective  $  n $-space in which a metric is defined by an [[Absolute|absolute]], consisting of an absolute cone  $  Q _ {0} $
 +
of index  $  k $
 +
with an  $  ( n - m ) $-vertex (an absolute plane  $  T _ {0} $)
 +
and an  $  ( n - m - 2 ) $-quadric (an absolute quadric  $  Q _ {1} $)
 +
of index  $  l $
 +
on this  $  ( n - m - 1 ) $-plane. Such a space is called a quasi-hyperbolic space of indices  $  k $
 +
and  $  l $,
 +
and is denoted by the symbol  $  {}  ^ {kl} S _ {n}  ^ {m} $,
 +
where  $  m < n $.  
 +
A quasi-hyperbolic space is a particular case of a [[Semi-hyperbolic space|semi-hyperbolic space]]. The quasi-hyperbolic space  $  {}  ^ {kl} S _ {n}  ^ {m} $
 +
is obtained from the hyperbolic space  $  {}  ^ {l} S _ {n} $
 +
by passing to the limit in such a way that the absolute of the hyperbolic space is transformed to the absolute of the quasi-hyperbolic space.
 +
 
 +
When  $  m = 0 $,  
 +
the cone $  Q _ {0} $
 +
is a pair of coincident planes that are the same as the plane $  T _ {0} $,  
 +
while the absolute of the space is the same as that of the [[Pseudo-Euclidean space|pseudo-Euclidean space]] $  {}  ^ {l} R _ {n} $.  
 +
When $  m = 1 $,  
 +
the cone $  Q _ {0} $
 +
is a pair of real planes; in particular, for $  {}  ^ {11} S _ {3}  ^ {1} $
 +
the plane $  T _ {0} $
 +
is the line of intersection of these two planes, while the quadric $  Q _ {1} $
 +
is a pair of points on $  T _ {0} $.  
 +
In the case $  m = n - 1 $,  
 +
the cone $  Q _ {0} $
 +
has point vertex and the absolute of $  {}  ^ {kl} S _ {n}  ^ {n-1} $
 +
is the same as that of the [[Co-pseudo-Euclidean space|co-pseudo-Euclidean space]] $  {}  ^ {l} R _ {n}  ^ {*} $.
  
 
Quasi-hyperbolic spaces are spaces of more general type in comparison to co-pseudo-Euclidean spaces.
 
Quasi-hyperbolic spaces are spaces of more general type in comparison to co-pseudo-Euclidean spaces.
  
The projective metric of the quasi-hyperbolic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653030.png" /> is defined in such a way that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653031.png" /> the metric of the pseudo-Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653032.png" /> is obtained, while when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653033.png" />, that of the co-pseudo-Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653034.png" /> is obtained.
+
The projective metric of the quasi-hyperbolic space $  {}  ^ {kl} S _ {n}  ^ {m} $
 +
is defined in such a way that when $  m = 0 $
 +
the metric of the pseudo-Euclidean space $  {}  ^ {l} R _ {n} $
 +
is obtained, while when $  m = n - 1 $,  
 +
that of the co-pseudo-Euclidean space $  {}  ^ {k} R _ {n}  ^ {*} $
 +
is obtained.
  
 
In a quasi-hyperbolic space lines of four types are distinguished: elliptic lines, intersecting the absolute cone in two conjugate-imaginary points; hyperbolic lines, intersecting the absolute cone in two real points; parabolic lines, passing through the vertex of the absolute cone; and isotropic lines, passing through the vertex of the absolute cone and tangent to it.
 
In a quasi-hyperbolic space lines of four types are distinguished: elliptic lines, intersecting the absolute cone in two conjugate-imaginary points; hyperbolic lines, intersecting the absolute cone in two real points; parabolic lines, passing through the vertex of the absolute cone; and isotropic lines, passing through the vertex of the absolute cone and tangent to it.
  
The distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653035.png" /> between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653037.png" /> is defined in case the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653038.png" /> does not intersect the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653039.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653040.png" /> by the formula
+
The distance $  \delta $
 
+
between two points $  X $
<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/q/q076/q076530/q07653041.png" /></td> </tr></table>
+
and $  Y $
 +
is defined in case the line $  X Y $
 +
does not intersect the $  ( n - m - 1 ) $-plane $  T _ {0} $
 +
by the formula
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653042.png" /> is the linear operator defining the scalar product in the pseudo-Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653043.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653044.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653046.png" /> are the vectors of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653048.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653049.png" /> is a real number. The distance between two points not lying on a parabolic line is equal to the distance between the projections of these points on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653050.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653051.png" /> in the direction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653052.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653053.png" />. In case the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653054.png" /> intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653055.png" />, the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653056.png" /> is calculated from the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653059.png" /> are the vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653061.png" /> in the pseudo-Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653062.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653064.png" /> being the linear operator defining the scalar product in this space.
+
$$
 +
\cos  ^ {2} 
 +
\frac \delta  \rho
 +
  = \
  
One takes as the angle between two planes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653065.png" /> the (normalized) distance between the two corresponding points in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653066.png" /> dual to it according to the duality principle of projective space. The coordinates of these points are numerically equal to the projective coordinates of the given planes. In case the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653067.png" />-plane of intersection of the two given planes intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653068.png" />, this angle is always zero, but in this case one applies a method of measurement analogous to that of measuring distance in the similar case. In particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653069.png" /> angles between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653070.png" />-planes are the angles between lines, and, depending on the position of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653071.png" />-plane relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653072.png" />, there are three possible types of geometry, namely Euclidean, pseudo-Euclidean and co-pseudo-Euclidean.
+
\frac{( \mathbf x  ^ {0} E _ {0} \mathbf y  ^ {0} )  ^ {2} }{( \mathbf x  ^ {0} E _ {0} \mathbf x  ^ {0} )
 +
( \mathbf y  ^ {0} E _ {0} \mathbf y  ^ {0} ) }
 +
,
 +
$$
  
The motions of a quasi-hyperbolic space are the collineations preserving distance between points and taking the absolute cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653073.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653074.png" />-vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653075.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653076.png" />-quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653077.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653078.png" /> into themselves. Motions are described by pseudo-orthogonal operators of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653079.png" />. In the quasi-hyperbolic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653080.png" />, which is self-dual, a co-motion is defined as a [[Correlation|correlation]] taking any two points to two planes the angle between which is proportional to the distance between the given points, and taking any two planes to two points the distance between which is proportional to the angle between the planes. The co-motions are described by pseudo-orthogonal operators of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653081.png" />. The motions form a Lie group, as do the motions and co-motions of a self-dual quasi-hyperbolic space.
+
where  $  E _ {0} $
 
+
is the linear operator defining the scalar product in the pseudo-Euclidean  $  ( m + 1 ) $-space  $  {}  ^ {k} R _ {m+1} $;
A quasi-hyperbolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653082.png" />-space with a projective elliptic metric on the lines, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653083.png" />, has a co-Euclidean metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653084.png" />-planes and a pseudo-Euclidean metric of index 1 in bundles of planes. A quasi-hyperbolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653085.png" />-space with a hyperbolic projective distance metric can be of two types: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653086.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653087.png" />, differing by their metrics in bundles of planes: in the first, a Euclidean, in the second, a pseudo-Euclidean metric of index 1. The metrics on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653088.png" />-planes coincide; it is a pseudo-Euclidean metric of index 1.
+
$  \mathbf x  ^ {0} = ( x  ^ {a} , a \leq  m ) $,
 
+
$  \mathbf y  ^ {0} = ( y  ^ {b} , b \leq  m ) $
The quasi-hyperbolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653089.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653090.png" /> can be interpreted as the group of motions of the pseudo-Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653091.png" />-plane of index 1. The manifold of hyperbolic lines on this quasi-hyperbolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653092.png" />-space can be interpreted as a pair of such pseudo-Euclidean planes. The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653094.png" /> are dual to each other, and can be interpreted on a complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076530/q07653095.png" />-plane.
+
are the vectors of the points  $  X $
 
+
and  $  Y $,  
====References====
+
and  $  \rho $
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.A. Rozenfel'd,   "Non-Euclidean spaces" , Moscow (1969) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.M. Yaglom,   B.A. Rozenfel'd,  E.U. Yasinskaya,  "Projective metrics" ''Russian Math. Surveys'' , '''19''' : 5 (1964) pp. 49–107 ''Uspekhi Mat. Nauk'' , '''19''' : 5 (1964pp. 51–113</TD></TR></table>
+
is a real number. The distance between two points not lying on a parabolic line is equal to the distance between the projections of these points on the $  m $-plane  $  \mathbf x  ^ {1} = 0 $
 +
in the direction of the $  ( n - m - 1 ) $-plane  $  T _ {0} $.  
 +
In case the line  $  X Y $
 +
intersects  $  T _ {0} $,
 +
the distance  $  d $
 +
is calculated from the difference  $  \mathbf a = \mathbf y  ^ {1} - \mathbf x  ^ {1} $,  
 +
where  $  \mathbf x  ^ {1} = ( x ^ {u} , u > m ) $,  
 +
$ \mathbf y ^ {1} = ( y ^ {v} , v > m ) $
 +
are the vectors of $ X $
 +
and $ Y $
 +
in the pseudo-Euclidean space $ R _ {n-m} $;
 +
$ d ( X , Y ) = \mathbf a E _ {1} \mathbf a $,
 +
$ E _ {1} $
 +
being the linear operator defining the scalar product in this space.
  
 +
One takes as the angle between two planes of  $  {}  ^ {kl} S _ {n}  ^ {m} $
 +
the (normalized) distance between the two corresponding points in the space  $  {}  ^ {lk} S _ {n}  ^ {n-m- 1} $
 +
dual to it according to the duality principle of projective space. The coordinates of these points are numerically equal to the projective coordinates of the given planes. In case the  $  ( n - 2 ) $-plane of intersection of the two given planes intersects  $  T _ {0} $,
 +
this angle is always zero, but in this case one applies a method of measurement analogous to that of measuring distance in the similar case. In particular, when  $  n = 2 $
 +
angles between  $  1 $-planes are the angles between lines, and, depending on the position of the  $  2 $-plane relative to  $  T _ {0} $,
 +
there are three possible types of geometry, namely Euclidean, pseudo-Euclidean and co-pseudo-Euclidean.
  
 +
The motions of a quasi-hyperbolic space are the collineations preserving distance between points and taking the absolute cone  $  Q _ {0} $,
 +
the  $  ( n - m - 1 ) $-vertex  $  T _ {0} $
 +
and the  $  ( n - m - 2 ) $-quadric  $  Q _ {1} $
 +
in  $  T _ {0} $
 +
into themselves. Motions are described by pseudo-orthogonal operators of index  $  l $.
 +
In the quasi-hyperbolic space  $  {}  ^ {ll} S _ {m+1}  ^ {m} $,
 +
which is self-dual, a co-motion is defined as a [[Correlation|correlation]] taking any two points to two planes the angle between which is proportional to the distance between the given points, and taking any two planes to two points the distance between which is proportional to the angle between the planes. The co-motions are described by pseudo-orthogonal operators of index  $  l $.
 +
The motions form a Lie group, as do the motions and co-motions of a self-dual quasi-hyperbolic space.
  
====Comments====
+
A quasi-hyperbolic  $  3 $-space with a projective elliptic metric on the lines,  $  {}  ^ {01} S _ {3}  ^ {1} $,
 +
has a co-Euclidean metric on  $  2 $-planes and a pseudo-Euclidean metric of index 1 in bundles of planes. A quasi-hyperbolic  $  3 $-space with a hyperbolic projective distance metric can be of two types:  $  {}  ^ {10} S _ {3}  ^ {1} $
 +
or  $  {}  ^ {11} S _ {3}  ^ {1} $,
 +
differing by their metrics in bundles of planes: in the first, a Euclidean, in the second, a pseudo-Euclidean metric of index 1. The metrics on the  $  2 $-planes coincide; it is a pseudo-Euclidean metric of index 1.
  
 +
The quasi-hyperbolic  $  3 $-space  $  {}  ^ {11} S _ {3}  ^ {1} $
 +
can be interpreted as the group of motions of the pseudo-Euclidean  $  2 $-plane of index 1. The manifold of hyperbolic lines on this quasi-hyperbolic  $  3 $-space can be interpreted as a pair of such pseudo-Euclidean planes. The spaces  $  {}  ^ {10} S _ {3}  ^ {1} $
 +
and  $  {}  ^ {01} S _ {3}  ^ {1} $
 +
are dual to each other, and can be interpreted on a complex  $  2 $-plane.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Giering,  "Vorlesungen über höhere Geometrie" , Vieweg  (1982)</TD></TR><TR><TD valign="top">[a2]</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">[1]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Yaglom,  B.A. Rozenfel'd,  E.U. Yasinskaya,  "Projective metrics"  ''Russian Math. Surveys'' , '''19''' :  5  (1964)  pp. 49–107  ''Uspekhi Mat. Nauk'' , '''19''' :  5  (1964)  pp. 51–113</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Giering,  "Vorlesungen über höhere Geometrie" , Vieweg  (1982)</TD></TR><TR><TD valign="top">[a2]</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>

Latest revision as of 14:58, 2 May 2023


A projective $ n $-space in which a metric is defined by an absolute, consisting of an absolute cone $ Q _ {0} $ of index $ k $ with an $ ( n - m ) $-vertex (an absolute plane $ T _ {0} $) and an $ ( n - m - 2 ) $-quadric (an absolute quadric $ Q _ {1} $) of index $ l $ on this $ ( n - m - 1 ) $-plane. Such a space is called a quasi-hyperbolic space of indices $ k $ and $ l $, and is denoted by the symbol $ {} ^ {kl} S _ {n} ^ {m} $, where $ m < n $. A quasi-hyperbolic space is a particular case of a semi-hyperbolic space. The quasi-hyperbolic space $ {} ^ {kl} S _ {n} ^ {m} $ is obtained from the hyperbolic space $ {} ^ {l} S _ {n} $ by passing to the limit in such a way that the absolute of the hyperbolic space is transformed to the absolute of the quasi-hyperbolic space.

When $ m = 0 $, the cone $ Q _ {0} $ is a pair of coincident planes that are the same as the plane $ T _ {0} $, while the absolute of the space is the same as that of the pseudo-Euclidean space $ {} ^ {l} R _ {n} $. When $ m = 1 $, the cone $ Q _ {0} $ is a pair of real planes; in particular, for $ {} ^ {11} S _ {3} ^ {1} $ the plane $ T _ {0} $ is the line of intersection of these two planes, while the quadric $ Q _ {1} $ is a pair of points on $ T _ {0} $. In the case $ m = n - 1 $, the cone $ Q _ {0} $ has point vertex and the absolute of $ {} ^ {kl} S _ {n} ^ {n-1} $ is the same as that of the co-pseudo-Euclidean space $ {} ^ {l} R _ {n} ^ {*} $.

Quasi-hyperbolic spaces are spaces of more general type in comparison to co-pseudo-Euclidean spaces.

The projective metric of the quasi-hyperbolic space $ {} ^ {kl} S _ {n} ^ {m} $ is defined in such a way that when $ m = 0 $ the metric of the pseudo-Euclidean space $ {} ^ {l} R _ {n} $ is obtained, while when $ m = n - 1 $, that of the co-pseudo-Euclidean space $ {} ^ {k} R _ {n} ^ {*} $ is obtained.

In a quasi-hyperbolic space lines of four types are distinguished: elliptic lines, intersecting the absolute cone in two conjugate-imaginary points; hyperbolic lines, intersecting the absolute cone in two real points; parabolic lines, passing through the vertex of the absolute cone; and isotropic lines, passing through the vertex of the absolute cone and tangent to it.

The distance $ \delta $ between two points $ X $ and $ Y $ is defined in case the line $ X Y $ does not intersect the $ ( n - m - 1 ) $-plane $ T _ {0} $ by the formula

$$ \cos ^ {2} \frac \delta \rho = \ \frac{( \mathbf x ^ {0} E _ {0} \mathbf y ^ {0} ) ^ {2} }{( \mathbf x ^ {0} E _ {0} \mathbf x ^ {0} ) ( \mathbf y ^ {0} E _ {0} \mathbf y ^ {0} ) } , $$

where $ E _ {0} $ is the linear operator defining the scalar product in the pseudo-Euclidean $ ( m + 1 ) $-space $ {} ^ {k} R _ {m+1} $; $ \mathbf x ^ {0} = ( x ^ {a} , a \leq m ) $, $ \mathbf y ^ {0} = ( y ^ {b} , b \leq m ) $ are the vectors of the points $ X $ and $ Y $, and $ \rho $ is a real number. The distance between two points not lying on a parabolic line is equal to the distance between the projections of these points on the $ m $-plane $ \mathbf x ^ {1} = 0 $ in the direction of the $ ( n - m - 1 ) $-plane $ T _ {0} $. In case the line $ X Y $ intersects $ T _ {0} $, the distance $ d $ is calculated from the difference $ \mathbf a = \mathbf y ^ {1} - \mathbf x ^ {1} $, where $ \mathbf x ^ {1} = ( x ^ {u} , u > m ) $, $ \mathbf y ^ {1} = ( y ^ {v} , v > m ) $ are the vectors of $ X $ and $ Y $ in the pseudo-Euclidean space $ R _ {n-m} $; $ d ( X , Y ) = \mathbf a E _ {1} \mathbf a $, $ E _ {1} $ being the linear operator defining the scalar product in this space.

One takes as the angle between two planes of $ {} ^ {kl} S _ {n} ^ {m} $ the (normalized) distance between the two corresponding points in the space $ {} ^ {lk} S _ {n} ^ {n-m- 1} $ dual to it according to the duality principle of projective space. The coordinates of these points are numerically equal to the projective coordinates of the given planes. In case the $ ( n - 2 ) $-plane of intersection of the two given planes intersects $ T _ {0} $, this angle is always zero, but in this case one applies a method of measurement analogous to that of measuring distance in the similar case. In particular, when $ n = 2 $ angles between $ 1 $-planes are the angles between lines, and, depending on the position of the $ 2 $-plane relative to $ T _ {0} $, there are three possible types of geometry, namely Euclidean, pseudo-Euclidean and co-pseudo-Euclidean.

The motions of a quasi-hyperbolic space are the collineations preserving distance between points and taking the absolute cone $ Q _ {0} $, the $ ( n - m - 1 ) $-vertex $ T _ {0} $ and the $ ( n - m - 2 ) $-quadric $ Q _ {1} $ in $ T _ {0} $ into themselves. Motions are described by pseudo-orthogonal operators of index $ l $. In the quasi-hyperbolic space $ {} ^ {ll} S _ {m+1} ^ {m} $, which is self-dual, a co-motion is defined as a correlation taking any two points to two planes the angle between which is proportional to the distance between the given points, and taking any two planes to two points the distance between which is proportional to the angle between the planes. The co-motions are described by pseudo-orthogonal operators of index $ l $. The motions form a Lie group, as do the motions and co-motions of a self-dual quasi-hyperbolic space.

A quasi-hyperbolic $ 3 $-space with a projective elliptic metric on the lines, $ {} ^ {01} S _ {3} ^ {1} $, has a co-Euclidean metric on $ 2 $-planes and a pseudo-Euclidean metric of index 1 in bundles of planes. A quasi-hyperbolic $ 3 $-space with a hyperbolic projective distance metric can be of two types: $ {} ^ {10} S _ {3} ^ {1} $ or $ {} ^ {11} S _ {3} ^ {1} $, differing by their metrics in bundles of planes: in the first, a Euclidean, in the second, a pseudo-Euclidean metric of index 1. The metrics on the $ 2 $-planes coincide; it is a pseudo-Euclidean metric of index 1.

The quasi-hyperbolic $ 3 $-space $ {} ^ {11} S _ {3} ^ {1} $ can be interpreted as the group of motions of the pseudo-Euclidean $ 2 $-plane of index 1. The manifold of hyperbolic lines on this quasi-hyperbolic $ 3 $-space can be interpreted as a pair of such pseudo-Euclidean planes. The spaces $ {} ^ {10} S _ {3} ^ {1} $ and $ {} ^ {01} S _ {3} ^ {1} $ are dual to each other, and can be interpreted on a complex $ 2 $-plane.

References

[1] B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)
[2] I.M. Yaglom, B.A. Rozenfel'd, E.U. Yasinskaya, "Projective metrics" Russian Math. Surveys , 19 : 5 (1964) pp. 49–107 Uspekhi Mat. Nauk , 19 : 5 (1964) pp. 51–113
[a1] O. Giering, "Vorlesungen über höhere Geometrie" , Vieweg (1982)
[a2] B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)
How to Cite This Entry:
Quasi-hyperbolic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-hyperbolic_space&oldid=13566
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article