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)
[a1]	B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)