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A projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q0764701.png" />-space whose projective metric is defined by an [[Absolute|absolute]] consisting of an imaginary cone (the absolute cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q0764702.png" />) with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q0764703.png" />-vertex (the absolute plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q0764704.png" />) together with an imaginary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q0764705.png" />-quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q0764706.png" /> on this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q0764707.png" />-plane (the absolute quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q0764708.png" />); it is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q0764709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647010.png" />. A quasi-elliptic space is of more general projective type in comparison with a Euclidean space and a [[Co-Euclidean space|co-Euclidean space]]; the metrics of the latter are obtained from those of the former. A quasi-elliptic space is a particular case of a [[Semi-elliptic space|semi-elliptic space]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647011.png" />, the absolute cone is a pair of coincident <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647012.png" />-planes coinciding with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647013.png" />-absolute plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647014.png" />, while the absolute coincides with the absolute of Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647015.png" />-space. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647016.png" />, the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647017.png" /> is a cone with a point vertex and the absolute in this case is the same as that of the co-Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647018.png" />-space. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647019.png" />, the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647020.png" /> is a pair of imaginary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647021.png" />-planes. In particular, the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647022.png" /> of the quasi-elliptic three-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647023.png" /> is a pair of imaginary two-planes, the line (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647024.png" />-plane) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647025.png" /> is the real line of their intersection, while the quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647026.png" /> is a pair of imaginary points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647027.png" />.
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The distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647028.png" /> between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647030.png" /> is defined in case the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647031.png" /> does not intersect the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647032.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647033.png" /> by the relation
+
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<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/q076470/q07647034.png" /></td> </tr></table>
+
A projective  $  n $-
 +
space whose projective metric is defined by an [[Absolute|absolute]] consisting of an imaginary cone (the absolute cone  $  Q _ {0} $)
 +
with an  $  ( n- m - 1 ) $-
 +
vertex (the absolute plane  $  T _ {0} $)
 +
together with an imaginary  $  ( n - m - 2 ) $-
 +
quadric  $  Q _ {1} $
 +
on this  $  ( n- m- 1) $-
 +
plane (the absolute quadric  $  Q _ {1} $);  
 +
it is denoted by the symbol  $  S _ {n}  ^ {m} $,
 +
$  m < n $.
 +
A quasi-elliptic space is of more general projective type in comparison with a Euclidean space and a [[Co-Euclidean space|co-Euclidean space]]; the metrics of the latter are obtained from those of the former. A quasi-elliptic space is a particular case of a [[Semi-elliptic space|semi-elliptic space]]. For  $  m = 0 $,
 +
the absolute cone is a pair of coincident  $  ( n- 1) $-
 +
planes coinciding with the  $  ( n- 1) $-
 +
absolute plane  $  T _ {0} $,
 +
while the absolute coincides with the absolute of Euclidean  $  n $-
 +
space. For  $  m= 1 $,
 +
the cone  $  Q _ {0} $
 +
is a cone with a point vertex and the absolute in this case is the same as that of the co-Euclidean  $  n $-
 +
space. When  $  m = 1 $,
 +
the cone  $  Q _ {0} $
 +
is a pair of imaginary  $  ( n - 1 ) $-
 +
planes. In particular, the cone  $  Q _ {0} $
 +
of the quasi-elliptic three-space  $  S _ {3}  ^ {1} $
 +
is a pair of imaginary two-planes, the line (the  $  1 $-
 +
plane)  $  T _ {0} $
 +
is the real line of their intersection, while the quadric  $  Q _ {1} $
 +
is a pair of imaginary points on  $  T _ {0} $.
 +
 
 +
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 relation
 +
 
 +
$$
 +
\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
 
where
  
<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/q076470/q07647035.png" /></td> </tr></table>
+
$$
 +
\mathbf x  ^ {0}  = ( x  ^ {a} , a \leq  m ) ,\ \
 +
\mathbf y  ^ {0= ( y  ^ {b} , b \leq  m )
 +
$$
  
are the vectors of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647038.png" /> is the linear operator defining the scalar product in the space of these vectors and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647039.png" /> is a real number; in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647040.png" /> intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647041.png" />, the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647042.png" /> between these points is defined by means of the distance between the vectors of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647044.png" />:
+
are the vectors of the points $  X $
 +
and $  Y $,  
 +
$  E _ {0} $
 +
is the linear operator defining the scalar product in the space of these vectors and $  \rho $
 +
is a real number; in case $  X Y $
 +
intersects $  T _ {0} $,  
 +
the distance $  d $
 +
between these points is defined by means of the distance between the vectors of the points $  X $
 +
and $  Y $:
  
<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/q076470/q07647045.png" /></td> </tr></table>
+
$$
 +
\mathbf x  = \mathbf y  ^ {1} - \mathbf x  ^ {1} ,
 +
$$
  
<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/q076470/q07647046.png" /></td> </tr></table>
+
$$
 +
\mathbf x  ^ {1}  = ( x  ^ {a} , a > m ) ,\  \mathbf y  ^ {1}  = ( y  ^ {b} , b > m ) ,
 +
$$
  
<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/q076470/q07647047.png" /></td> </tr></table>
+
$$
 +
d  ^ {2}  = \mathbf a E _ {1} \mathbf a ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647048.png" /> is the linear operator defining the scalar product in the space of these vectors.
+
where $  E _ {1} $
 +
is the linear operator defining the scalar product in the space of these vectors.
  
The angle between two planes whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647049.png" />-plane of intersection does not intersect the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647050.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647051.png" /> is defined as the (normalized) distance between the corresponding points in the dual quasi-elliptic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647052.png" />, in which the coordinates are numerically equal or proportional to the projective coordinates of the planes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647053.png" />. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647054.png" />-plane of intersection of two given planes intersects the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647055.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647056.png" />, then the angle between the planes is in this case again defined by the numerical distance. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647057.png" /> the angles between the planes are the angles between the lines.
+
The angle between two planes whose $  ( n- 2 ) $-
 +
plane of intersection does not intersect the $  ( n - m - 1 ) $-
 +
plane $  T _ {0} $
 +
is defined as the (normalized) distance between the corresponding points in the dual quasi-elliptic space $  S _ {n}  ^ {n-} m- 1 $,  
 +
in which the coordinates are numerically equal or proportional to the projective coordinates of the planes in $  S _ {n}  ^ {m} $.  
 +
If the $  ( n - 2 ) $-
 +
plane of intersection of two given planes intersects the $  ( n- m- 1 ) $-
 +
plane $  T _ {0} $,  
 +
then the angle between the planes is in this case again defined by the numerical distance. When $  n = 2 $
 +
the angles between the planes are the angles between the lines.
  
The motions of the quasi-elliptic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647058.png" /> are the collineations of this space that take the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647059.png" /> into the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647060.png" /> and the quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647061.png" /> into itself. The group of motions is a Lie group and the motions are described by orthogonal operators. In the quasi-elliptic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647062.png" />, which is self-dual, co-motions are defined as the correlations that take each pair of points into two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647063.png" />-planes the angle between which is proportional to the distance between the points, and each pair of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647064.png" />-planes into two points the distance between which is proportional to the angle between the planes. The motions and co-motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647065.png" /> form a group, which is a Lie group. The geometry of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647066.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647067.png" /> is Euclidean geometry, while the geometry of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647068.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647069.png" /> is the same as that of the co-Euclidean plane.
+
The motions of the quasi-elliptic space $  S _ {n}  ^ {m} $
 +
are the collineations of this space that take the cone $  Q _ {0} $
 +
into the plane $  T _ {0} $
 +
and the quadric $  Q _ {1} $
 +
into itself. The group of motions is a Lie group and the motions are described by orthogonal operators. In the quasi-elliptic space $  S _ {2m+} 1  ^ {m} $,  
 +
which is self-dual, co-motions are defined as the correlations that take each pair of points into two $  2m $-
 +
planes the angle between which is proportional to the distance between the points, and each pair of $  2 m $-
 +
planes into two points the distance between which is proportional to the angle between the planes. The motions and co-motions of $  S _ {2m+} 1  ^ {m} $
 +
form a group, which is a Lie group. The geometry of the $  2 $-
 +
plane $  S _ {2}  ^ {0} $
 +
is Euclidean geometry, while the geometry of the $  2 $-
 +
plane $  S _ {2}  ^ {1} $
 +
is the same as that of the co-Euclidean plane.
  
The geometry of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647070.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647071.png" /> is defined by an elliptic projective metric on lines that is co-Euclidean on planes and Euclidean in bundles of planes. The geometry of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647072.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647073.png" /> is Euclidean, while the geometry of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647074.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647075.png" /> is the same as that of the co-Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647076.png" />-space. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647077.png" /> with radius of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647078.png" /> is isometric to the connected group of motions of the Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647079.png" />-space with a specially introduced metric. The connected group of motions of the quasi-Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647080.png" /> is isomorphic to the direct product of two connected groups of motions of the Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076470/q07647081.png" />-plane.
+
The geometry of the $  3 $-
 +
space $  S _ {3}  ^ {1} $
 +
is defined by an elliptic projective metric on lines that is co-Euclidean on planes and Euclidean in bundles of planes. The geometry of the $  3 $-
 +
space $  S _ {3}  ^ {0} $
 +
is Euclidean, while the geometry of the $  3 $-
 +
space $  S _ {3}  ^ {2} $
 +
is the same as that of the co-Euclidean $  3 $-
 +
space. The space $  S _ {3}  ^ {1} $
 +
with radius of curvature $  1 / 2 $
 +
is isometric to the connected group of motions of the Euclidean $  2 $-
 +
space with a specially introduced metric. The connected group of motions of the quasi-Euclidean space $  S _ {3}  ^ {1} $
 +
is isomorphic to the direct product of two connected groups of motions of the Euclidean $  2 $-
 +
plane.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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">  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)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Giering,  "Vorlesungen über höhere Geometrie" , Vieweg  (1982)</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)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Giering,  "Vorlesungen über höhere Geometrie" , Vieweg  (1982)</TD></TR></table>

Latest revision as of 08:09, 6 June 2020


A projective $ n $- space whose projective metric is defined by an absolute consisting of an imaginary cone (the absolute cone $ Q _ {0} $) with an $ ( n- m - 1 ) $- vertex (the absolute plane $ T _ {0} $) together with an imaginary $ ( n - m - 2 ) $- quadric $ Q _ {1} $ on this $ ( n- m- 1) $- plane (the absolute quadric $ Q _ {1} $); it is denoted by the symbol $ S _ {n} ^ {m} $, $ m < n $. A quasi-elliptic space is of more general projective type in comparison with a Euclidean space and a co-Euclidean space; the metrics of the latter are obtained from those of the former. A quasi-elliptic space is a particular case of a semi-elliptic space. For $ m = 0 $, the absolute cone is a pair of coincident $ ( n- 1) $- planes coinciding with the $ ( n- 1) $- absolute plane $ T _ {0} $, while the absolute coincides with the absolute of Euclidean $ n $- space. For $ m= 1 $, the cone $ Q _ {0} $ is a cone with a point vertex and the absolute in this case is the same as that of the co-Euclidean $ n $- space. When $ m = 1 $, the cone $ Q _ {0} $ is a pair of imaginary $ ( n - 1 ) $- planes. In particular, the cone $ Q _ {0} $ of the quasi-elliptic three-space $ S _ {3} ^ {1} $ is a pair of imaginary two-planes, the line (the $ 1 $- plane) $ T _ {0} $ is the real line of their intersection, while the quadric $ Q _ {1} $ is a pair of imaginary points on $ T _ {0} $.

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 relation

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

where

$$ \mathbf x ^ {0} = ( x ^ {a} , a \leq m ) ,\ \ \mathbf y ^ {0} = ( y ^ {b} , b \leq m ) $$

are the vectors of the points $ X $ and $ Y $, $ E _ {0} $ is the linear operator defining the scalar product in the space of these vectors and $ \rho $ is a real number; in case $ X Y $ intersects $ T _ {0} $, the distance $ d $ between these points is defined by means of the distance between the vectors of the points $ X $ and $ Y $:

$$ \mathbf x = \mathbf y ^ {1} - \mathbf x ^ {1} , $$

$$ \mathbf x ^ {1} = ( x ^ {a} , a > m ) ,\ \mathbf y ^ {1} = ( y ^ {b} , b > m ) , $$

$$ d ^ {2} = \mathbf a E _ {1} \mathbf a , $$

where $ E _ {1} $ is the linear operator defining the scalar product in the space of these vectors.

The angle between two planes whose $ ( n- 2 ) $- plane of intersection does not intersect the $ ( n - m - 1 ) $- plane $ T _ {0} $ is defined as the (normalized) distance between the corresponding points in the dual quasi-elliptic space $ S _ {n} ^ {n-} m- 1 $, in which the coordinates are numerically equal or proportional to the projective coordinates of the planes in $ S _ {n} ^ {m} $. If the $ ( n - 2 ) $- plane of intersection of two given planes intersects the $ ( n- m- 1 ) $- plane $ T _ {0} $, then the angle between the planes is in this case again defined by the numerical distance. When $ n = 2 $ the angles between the planes are the angles between the lines.

The motions of the quasi-elliptic space $ S _ {n} ^ {m} $ are the collineations of this space that take the cone $ Q _ {0} $ into the plane $ T _ {0} $ and the quadric $ Q _ {1} $ into itself. The group of motions is a Lie group and the motions are described by orthogonal operators. In the quasi-elliptic space $ S _ {2m+} 1 ^ {m} $, which is self-dual, co-motions are defined as the correlations that take each pair of points into two $ 2m $- planes the angle between which is proportional to the distance between the points, and each pair of $ 2 m $- planes into two points the distance between which is proportional to the angle between the planes. The motions and co-motions of $ S _ {2m+} 1 ^ {m} $ form a group, which is a Lie group. The geometry of the $ 2 $- plane $ S _ {2} ^ {0} $ is Euclidean geometry, while the geometry of the $ 2 $- plane $ S _ {2} ^ {1} $ is the same as that of the co-Euclidean plane.

The geometry of the $ 3 $- space $ S _ {3} ^ {1} $ is defined by an elliptic projective metric on lines that is co-Euclidean on planes and Euclidean in bundles of planes. The geometry of the $ 3 $- space $ S _ {3} ^ {0} $ is Euclidean, while the geometry of the $ 3 $- space $ S _ {3} ^ {2} $ is the same as that of the co-Euclidean $ 3 $- space. The space $ S _ {3} ^ {1} $ with radius of curvature $ 1 / 2 $ is isometric to the connected group of motions of the Euclidean $ 2 $- space with a specially introduced metric. The connected group of motions of the quasi-Euclidean space $ S _ {3} ^ {1} $ is isomorphic to the direct product of two connected groups of motions of the Euclidean $ 2 $- plane.

References

[1] 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) (Translated from Russian)
[a2] O. Giering, "Vorlesungen über höhere Geometrie" , Vieweg (1982)
How to Cite This Entry:
Quasi-elliptic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-elliptic_space&oldid=48380
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article