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A projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s0840801.png" />-space in which the metric is defined by a given absolute, which is the aggregate of an imaginary quadratic cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s0840802.png" /> with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s0840803.png" />-flat vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s0840804.png" />, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s0840805.png" />-imaginary cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s0840806.png" /> with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s0840807.png" />-flat vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s0840808.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s0840809.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408010.png" />, etc., up to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408011.png" />-imaginary cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408012.png" /> with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408013.png" />-flat vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408014.png" /> and a non-degenerate imaginary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408015.png" />-quadratic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408016.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408017.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408019.png" />. The indices of the cones <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408021.png" />, are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408023.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408024.png" />. A semi-elliptic space is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408025.png" />.
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A projective $n$-space in which the metric is defined by a given absolute, which is the aggregate of an imaginary quadratic cone $Q_0$ with an $(n-m_0-1)$-flat vertex $T_0$, an $(n-m_0-2)$-imaginary cone $Q_1$ with an $(n-m_1-1)$-flat vertex $T_1$ in the $(n-m_0-1)$-plane $T_0$, etc., up to an $(n-m_{r-1}-2)$-imaginary cone $Q_{r-1}$ with an $(n-m_{r-1}-2)$-flat vertex $T_{r-1}$ and a non-degenerate imaginary $(n-m_{r-1}-2)$-quadratic $Q_r$ in the $(n-m_{r-1}-1)$-plane $T_{r-1}$, $0\leq m_0<m_1<\ldots<m_{r-1}<n$. The indices of the cones $Q_k$, $k=0,\ldots,r-1$, are: $l_a=m_0-m_{a-1}$, $0<a<r$; $l_r=n-m_{r-1}$. A semi-elliptic space is denoted by $S_n^{m_0\ldots m_{r-1}}$.
  
In case the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408026.png" /> is a pair of merging planes coinciding with the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408027.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408028.png" />), the space with the improper plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408029.png" /> is called the [[Semi-Euclidean space|semi-Euclidean space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408030.png" />.
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In case the cone $Q_0$ is a pair of merging planes coinciding with the plane $T_0$ (for $m_0=0$), the space with the improper plane $T_0$ is called the [[Semi-Euclidean space|semi-Euclidean space]] $R_n^{m_1\ldots m_{r-1}}$.
  
The distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408032.png" /> is defined according to the position of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408033.png" /> with respect to the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408034.png" />. If, in particular, the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408035.png" /> does not intersect the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408036.png" />, then the distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408038.png" /> is defined in terms of the scalar product, analogously to the distance in a [[Quasi-elliptic space|quasi-elliptic space]]. If, however, the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408039.png" /> intersects the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408040.png" /> but does not intersect the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408041.png" />, or intersects the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408042.png" /> but does not intersect the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408043.png" />, the distance between the points is defined using the scalar square of the difference of the corresponding vectors of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408045.png" />.
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The distance between two points $X$ and $Y$ is defined according to the position of the straight line $XY$ with respect to the planes $T_0,\ldots,T_{r-1}$. If, in particular, the line $XY$ does not intersect the plane $T_0$, then the distance between the points $X$ and $Y$ is defined in terms of the scalar product, analogously to the distance in a [[Quasi-elliptic space|quasi-elliptic space]]. If, however, the line $XY$ intersects the plane $T_0$ but does not intersect the plane $T_1$, or intersects the plane $T_{a-1}$ but does not intersect the plane $T_a$, the distance between the points is defined using the scalar square of the difference of the corresponding vectors of the points $X$ and $Y$.
  
 
According to the position with respect to the planes of the absolute in a semi-elliptic space, one distinguishes four types of straight lines.
 
According to the position with respect to the planes of the absolute in a semi-elliptic space, one distinguishes four types of straight lines.
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The angles between planes in a semi-elliptic space are defined analogously to angles between planes in a quasi-elliptic space, that is, by using distances in the dual space.
 
The angles between planes in a semi-elliptic space are defined analogously to angles between planes in a quasi-elliptic space, that is, by using distances in the dual space.
  
A projective metric in a semi-elliptic space is a metric of a very general type. A particular case of the metric in a semi-elliptic space is, for example, the metric of a quasi-elliptic space. In particular, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408046.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408047.png" /> coincides with the Euclidean and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408048.png" /> with the co-Euclidean plane; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408049.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408050.png" /> with the quasi-elliptic and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408051.png" /> with the Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408052.png" />-space; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408053.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408054.png" /> is Galilean, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408055.png" /> is a flag space, etc. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408056.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408057.png" /> corresponds by the duality principle to the Galilean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408058.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408059.png" /> and is called the co-Galilean space. (The absolute of a co-Galilean space consists of a pair of imaginary planes (the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408060.png" />) and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408061.png" /> on the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408062.png" /> of intersection of these planes.)
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A projective metric in a semi-elliptic space is a metric of a very general type. A particular case of the metric in a semi-elliptic space is, for example, the metric of a quasi-elliptic space. In particular, the $2$-plane $S_2^0$ coincides with the Euclidean and $S_2^1$ with the co-Euclidean plane; the $3$-space $S_3^1$ with the quasi-elliptic and $S_3^0$ with the Euclidean $3$-space; the $3$-space $S_3^{01}$ is Galilean, $S_3^{012}$ is a flag space, etc. The $3$-space $S_3^{12}$ corresponds by the duality principle to the Galilean $3$-space $\Gamma_3$ and is called the co-Galilean space. (The absolute of a co-Galilean space consists of a pair of imaginary planes (the cone $Q_0$) and a point $T_1$ on the straight line $T_0$ of intersection of these planes.)
  
The motions of a semi-elliptic space are the collineations of it taking the absolute into itself. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084080/s08408064.png" />, the semi-elliptic space is dual to itself, and has co-motions defined in it analogously to co-motions in a quasi-elliptic space.
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The motions of a semi-elliptic space are the collineations of it taking the absolute into itself. In the case $m_a=n-m_{r-a-1}-1$, $l_a=l_{r-a}$, the semi-elliptic space is dual to itself, and has co-motions defined in it analogously to co-motions in a quasi-elliptic space.
  
 
The motions, and the motions and co-motions form Lie groups. The motions (as well as the co-motions) are described by orthogonal operators.
 
The motions, and the motions and co-motions form Lie groups. The motions (as well as the co-motions) are described by orthogonal operators.

Revision as of 18:50, 17 August 2014

A projective $n$-space in which the metric is defined by a given absolute, which is the aggregate of an imaginary quadratic cone $Q_0$ with an $(n-m_0-1)$-flat vertex $T_0$, an $(n-m_0-2)$-imaginary cone $Q_1$ with an $(n-m_1-1)$-flat vertex $T_1$ in the $(n-m_0-1)$-plane $T_0$, etc., up to an $(n-m_{r-1}-2)$-imaginary cone $Q_{r-1}$ with an $(n-m_{r-1}-2)$-flat vertex $T_{r-1}$ and a non-degenerate imaginary $(n-m_{r-1}-2)$-quadratic $Q_r$ in the $(n-m_{r-1}-1)$-plane $T_{r-1}$, $0\leq m_0<m_1<\ldots<m_{r-1}<n$. The indices of the cones $Q_k$, $k=0,\ldots,r-1$, are: $l_a=m_0-m_{a-1}$, $0<a<r$; $l_r=n-m_{r-1}$. A semi-elliptic space is denoted by $S_n^{m_0\ldots m_{r-1}}$.

In case the cone $Q_0$ is a pair of merging planes coinciding with the plane $T_0$ (for $m_0=0$), the space with the improper plane $T_0$ is called the semi-Euclidean space $R_n^{m_1\ldots m_{r-1}}$.

The distance between two points $X$ and $Y$ is defined according to the position of the straight line $XY$ with respect to the planes $T_0,\ldots,T_{r-1}$. If, in particular, the line $XY$ does not intersect the plane $T_0$, then the distance between the points $X$ and $Y$ is defined in terms of the scalar product, analogously to the distance in a quasi-elliptic space. If, however, the line $XY$ intersects the plane $T_0$ but does not intersect the plane $T_1$, or intersects the plane $T_{a-1}$ but does not intersect the plane $T_a$, the distance between the points is defined using the scalar square of the difference of the corresponding vectors of the points $X$ and $Y$.

According to the position with respect to the planes of the absolute in a semi-elliptic space, one distinguishes four types of straight lines.

The angles between planes in a semi-elliptic space are defined analogously to angles between planes in a quasi-elliptic space, that is, by using distances in the dual space.

A projective metric in a semi-elliptic space is a metric of a very general type. A particular case of the metric in a semi-elliptic space is, for example, the metric of a quasi-elliptic space. In particular, the $2$-plane $S_2^0$ coincides with the Euclidean and $S_2^1$ with the co-Euclidean plane; the $3$-space $S_3^1$ with the quasi-elliptic and $S_3^0$ with the Euclidean $3$-space; the $3$-space $S_3^{01}$ is Galilean, $S_3^{012}$ is a flag space, etc. The $3$-space $S_3^{12}$ corresponds by the duality principle to the Galilean $3$-space $\Gamma_3$ and is called the co-Galilean space. (The absolute of a co-Galilean space consists of a pair of imaginary planes (the cone $Q_0$) and a point $T_1$ on the straight line $T_0$ of intersection of these planes.)

The motions of a semi-elliptic space are the collineations of it taking the absolute into itself. In the case $m_a=n-m_{r-a-1}-1$, $l_a=l_{r-a}$, the semi-elliptic space is dual to itself, and has co-motions defined in it analogously to co-motions in a quasi-elliptic space.

The motions, and the motions and co-motions form Lie groups. The motions (as well as the co-motions) are described by orthogonal operators.

A semi-elliptic space is a semi-Riemannian space.

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)
How to Cite This Entry:
Semi-elliptic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-elliptic_space&oldid=18405
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article