View source for Quasi-elliptic space

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" />.

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

<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>

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>

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" />:

<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>

<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>

<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>

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.

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 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 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.

====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>



====Comments====


====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>