View source for Semi-hyperbolic space

A projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s0841701.png" />-space in which the metric is defined by a given absolute consisting of the following collection: a second-order real cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s0841702.png" /> of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s0841703.png" /> with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s0841704.png" />-plane vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s0841705.png" />; a real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s0841706.png" />-cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s0841707.png" /> of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s0841708.png" /> with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s0841709.png" />-plane vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417010.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417011.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417012.png" />;<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417013.png" />; a real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417014.png" />-cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417015.png" /> of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417016.png" /> with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417017.png" />-plane vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417018.png" />; and a non-degenerate real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417019.png" />-quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417020.png" /> of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417021.png" /> in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417022.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417023.png" />. This is the definition of a semi-hyperbolic space with indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417024.png" />; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417025.png" />.

If the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417026.png" /> is a pair of merging planes, both identical with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417027.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417028.png" />), the semi-hyperbolic plane with the improper plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417029.png" /> is called a [[Semi-Euclidean space|semi-Euclidean space]]:

<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/s/s084/s084170/s08417030.png" /></td> </tr></table>

The distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417032.png" /> is defined as a function of the position of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417033.png" /> relative to the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417034.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417035.png" /> does not intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417036.png" />, the distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417038.png" /> is defined through a scalar product, in analogy with the appropriate definition in a [[Quasi-hyperbolic space|quasi-hyperbolic space]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417039.png" /> intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417040.png" /> but does not intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417041.png" />, or it intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417042.png" /> but does not intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417043.png" />, the distance between the points is defined as the scalar product with itself of the distance between the vectors of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417045.png" />.

Depending on the position of the absolute relative to the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417046.png" /> one distinguishes four types of straight lines of different orders: elliptic, hyperbolic, isotropic, and parabolic.

The angles between the planes in a semi-hyperbolic space are defined analogous to the angles between the planes in a quasi-hyperbolic space, i.e. using distance in the dual space.

A projective metric in a semi-hyperbolic space is a metric of the most general form. A particular case of such a metric is a metric of a quasi-hyperbolic space. In particular, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417047.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417048.png" /> is identical with the pseudo-Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417049.png" />, the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417050.png" /> — with the co-pseudo-Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417051.png" />; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417052.png" />-spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417054.png" /> coincide with the quasi-hyperbolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417055.png" />-space, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417056.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417057.png" /> — with the co-pseudo-Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417058.png" />, etc. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417059.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417060.png" /> is dual to the pseudo-Galilean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417061.png" />, it is known as a co-pseudo-Galilean space; its absolute consists of pairs of real planes (a cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417062.png" />) and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417063.png" /> on the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417064.png" /> in which these planes intersect.

The motions of a semi-hyperbolic space are defined as collineations of the space which map the absolute into itself. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084170/s08417066.png" />, a semi-hyperbolic space is dual to itself. It is then possible to define co-motions, the definition being analogous to that of co-motions in a self-dual quasi-hyperbolic space. The group of motions and the group of motions and co-motions are Lie groups. The motions (and co-motions) of a semi-hyperbolic space are described by pseudo-orthogonal operators with indices determined by the indices of the space.

A semi-hyperbolic space is a [[Semi-Riemannian space|semi-Riemannian space]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.M.Y. Sommerville,   ''Proc. Edinburgh Math. Soc.'' , '''28'''  (1910)  pp. 25–41</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Rozenfel'd,   "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>



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====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></table>