# Semi-hyperbolic space

A projective $n$- space in which the metric is defined by a given absolute consisting of the following collection: a second-order real cone $Q _ {0}$ of index $l _ {0}$ with an $( n - m _ {0} - 1 )$- plane vertex $T _ {0}$; a real $( n - m _ {0} - 2 )$- cone $Q _ {1}$ of index $l _ {1}$ with an $( n - m _ {1} - 1 )$- plane vertex $T _ {1}$ in the $( n - m _ {0} - 1 )$- plane $T _ {0}$; $\dots$; a real $( n - m _ {r-} 2 - 2 )$- cone $Q _ {r-} 1$ of index $l _ {r-} 1$ with an $( n - m _ {r-} 1 - 1 )$- plane vertex $T _ {r-} 1$; and a non-degenerate real $( n - m _ {r-} 1 - 2 )$- quadric $Q _ {2}$ of index $l _ {2}$ in the plane $T _ {r-} 1$; $0 \leq m _ {0} < m _ {1} < \dots < m _ {r-} 1 < n$. This is the definition of a semi-hyperbolic space with indices $l _ {0} \dots l _ {r}$; it is denoted by ${} ^ {l _ {0} {} \dots l _ {r} } S _ {n} ^ {m _ {0} \dots m _ {r-} 1 }$.

If the cone $Q _ {0}$ is a pair of merging planes, both identical with $T _ {0}$( for $m _ {0} = 0$), the semi-hyperbolic plane with the improper plane $T _ {0}$ is called a semi-Euclidean space:

$${} ^ {l _ {1} \dots l _ {r} } R _ {n} ^ {m _ {1} \dots m _ {r-} 1 } .$$

The distance between two points $X$ and $Y$ is defined as a function of the position of the straight line $X Y$ relative to the planes $T _ {0} \dots T _ {r-} 1$. In particular, if $X X$ does not intersect $T _ {0}$, the distance between $X$ and $Y$ is defined through a scalar product, in analogy with the appropriate definition in a quasi-hyperbolic space. If $X Y$ intersects $T _ {0}$ but does not intersect $T _ {1}$, or it intersects $T _ {a-} 1$ but does not intersect $T _ {a}$, the distance between the points is defined as the scalar product with itself of the distance between the vectors of the points $X$ and $Y$.

Depending on the position of the absolute relative to the planes $T _ {0} \dots T _ {a} \dots$ 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 $2$- plane ${} ^ {01} S _ {2} ^ {0}$ is identical with the pseudo-Euclidean space ${} ^ {1} R _ {2}$, the plane ${} ^ {10} S _ {2} ^ {1}$— with the co-pseudo-Euclidean space ${} ^ {1} R _ {2} ^ {*}$; the $3$- spaces ${} ^ {11} S _ {3} ^ {1}$ and ${} ^ {10} S _ {3} ^ {1}$ coincide with the quasi-hyperbolic $3$- space, the $3$- space ${} ^ {10} S _ {3} ^ {2}$— with the co-pseudo-Euclidean space ${} ^ {1} R _ {3} ^ {*}$, etc. The $3$- space ${} ^ {100} S _ {3} ^ {12}$ is dual to the pseudo-Galilean space ${} ^ {1} \Gamma _ {3}$, it is known as a co-pseudo-Galilean space; its absolute consists of pairs of real planes (a cone $Q _ {0}$) and a point $T _ {1}$ on the straight line $T _ {0}$ 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 $m _ {a} = n- m _ {r-} a- 1 - 1$ and $l _ {a} = l _ {r-} a$, 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.

#### References

 [1] D.M.Y. Sommerville, Proc. Edinburgh Math. Soc. , 28 (1910) pp. 25–41 [2] B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)