Semi-elliptic space

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 $T_{r-1}$, $0\leq m_0<m_1<\dots<m_{r-1}<n$. The indices of the cones $Q_k$, $k=0,\dots,r-1$, are: $l_a=m_0-m_{a-1}$, $0<a<r$; $l_r=n-m_{r-1}$. A semi-elliptic space is denoted by $S_n^{m_0\dots m_{r-1}}$.

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 $R_n^{m_1\dots m_{r-1}}$.

The distance between two points $X$ and $Y$ is defined according to the position of the straight line $XY$ with respect to the planes $T_0,\dots,T_{r-1}$. 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)
[a1]	B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)