Quasi-hyperbolic space

A projective $ n $-space in which a metric is defined by an absolute, consisting of an absolute cone $ Q _ {0} $ of index $ k $ with an $ ( n - m ) $-vertex (an absolute plane $ T _ {0} $) and an $ ( n - m - 2 ) $-quadric (an absolute quadric $ Q _ {1} $) of index $ l $ on this $ ( n - m - 1 ) $-plane. Such a space is called a quasi-hyperbolic space of indices $ k $ and $ l $, and is denoted by the symbol $ {} ^ {kl} S _ {n} ^ {m} $, where $ m < n $. A quasi-hyperbolic space is a particular case of a semi-hyperbolic space. The quasi-hyperbolic space $ {} ^ {kl} S _ {n} ^ {m} $ is obtained from the hyperbolic space $ {} ^ {l} S _ {n} $ by passing to the limit in such a way that the absolute of the hyperbolic space is transformed to the absolute of the quasi-hyperbolic space.

When $ m = 0 $, the cone $ Q _ {0} $ is a pair of coincident planes that are the same as the plane $ T _ {0} $, while the absolute of the space is the same as that of the pseudo-Euclidean space $ {} ^ {l} R _ {n} $. When $ m = 1 $, the cone $ Q _ {0} $ is a pair of real planes; in particular, for $ {} ^ {11} S _ {3} ^ {1} $ the plane $ T _ {0} $ is the line of intersection of these two planes, while the quadric $ Q _ {1} $ is a pair of points on $ T _ {0} $. In the case $ m = n - 1 $, the cone $ Q _ {0} $ has point vertex and the absolute of $ {} ^ {kl} S _ {n} ^ {n-1} $ is the same as that of the co-pseudo-Euclidean space $ {} ^ {l} R _ {n} ^ {*} $.

Quasi-hyperbolic spaces are spaces of more general type in comparison to co-pseudo-Euclidean spaces.

The projective metric of the quasi-hyperbolic space $ {} ^ {kl} S _ {n} ^ {m} $ is defined in such a way that when $ m = 0 $ the metric of the pseudo-Euclidean space $ {} ^ {l} R _ {n} $ is obtained, while when $ m = n - 1 $, that of the co-pseudo-Euclidean space $ {} ^ {k} R _ {n} ^ {*} $ is obtained.

In a quasi-hyperbolic space lines of four types are distinguished: elliptic lines, intersecting the absolute cone in two conjugate-imaginary points; hyperbolic lines, intersecting the absolute cone in two real points; parabolic lines, passing through the vertex of the absolute cone; and isotropic lines, passing through the vertex of the absolute cone and tangent to it.

The distance $ \delta $ between two points $ X $ and $ Y $ is defined in case the line $ X Y $ does not intersect the $ ( n - m - 1 ) $-plane $ T _ {0} $ by the formula

$$ \cos ^ {2} \frac \delta \rho = \ \frac{( \mathbf x ^ {0} E _ {0} \mathbf y ^ {0} ) ^ {2} }{( \mathbf x ^ {0} E _ {0} \mathbf x ^ {0} ) ( \mathbf y ^ {0} E _ {0} \mathbf y ^ {0} ) } , $$

where $ E _ {0} $ is the linear operator defining the scalar product in the pseudo-Euclidean $ ( m + 1 ) $-space $ {} ^ {k} R _ {m+1} $; $ \mathbf x ^ {0} = ( x ^ {a} , a \leq m ) $, $ \mathbf y ^ {0} = ( y ^ {b} , b \leq m ) $ are the vectors of the points $ X $ and $ Y $, and $ \rho $ is a real number. The distance between two points not lying on a parabolic line is equal to the distance between the projections of these points on the $ m $-plane $ \mathbf x ^ {1} = 0 $ in the direction of the $ ( n - m - 1 ) $-plane $ T _ {0} $. In case the line $ X Y $ intersects $ T _ {0} $, the distance $ d $ is calculated from the difference $ \mathbf a = \mathbf y ^ {1} - \mathbf x ^ {1} $, where $ \mathbf x ^ {1} = ( x ^ {u} , u > m ) $, $ \mathbf y ^ {1} = ( y ^ {v} , v > m ) $ are the vectors of $ X $ and $ Y $ in the pseudo-Euclidean space $ R _ {n-m} $; $ d ( X , Y ) = \mathbf a E _ {1} \mathbf a $, $ E _ {1} $ being the linear operator defining the scalar product in this space.

One takes as the angle between two planes of $ {} ^ {kl} S _ {n} ^ {m} $ the (normalized) distance between the two corresponding points in the space $ {} ^ {lk} S _ {n} ^ {n-m- 1} $ dual to it according to the duality principle of projective space. The coordinates of these points are numerically equal to the projective coordinates of the given planes. In case the $ ( n - 2 ) $-plane of intersection of the two given planes intersects $ T _ {0} $, this angle is always zero, but in this case one applies a method of measurement analogous to that of measuring distance in the similar case. In particular, when $ n = 2 $ angles between $ 1 $-planes are the angles between lines, and, depending on the position of the $ 2 $-plane relative to $ T _ {0} $, there are three possible types of geometry, namely Euclidean, pseudo-Euclidean and co-pseudo-Euclidean.

The motions of a quasi-hyperbolic space are the collineations preserving distance between points and taking the absolute cone $ Q _ {0} $, the $ ( n - m - 1 ) $-vertex $ T _ {0} $ and the $ ( n - m - 2 ) $-quadric $ Q _ {1} $ in $ T _ {0} $ into themselves. Motions are described by pseudo-orthogonal operators of index $ l $. In the quasi-hyperbolic space $ {} ^ {ll} S _ {m+1} ^ {m} $, which is self-dual, a co-motion is defined as a correlation taking any two points to two planes the angle between which is proportional to the distance between the given points, and taking any two planes to two points the distance between which is proportional to the angle between the planes. The co-motions are described by pseudo-orthogonal operators of index $ l $. The motions form a Lie group, as do the motions and co-motions of a self-dual quasi-hyperbolic space.

A quasi-hyperbolic $ 3 $-space with a projective elliptic metric on the lines, $ {} ^ {01} S _ {3} ^ {1} $, has a co-Euclidean metric on $ 2 $-planes and a pseudo-Euclidean metric of index 1 in bundles of planes. A quasi-hyperbolic $ 3 $-space with a hyperbolic projective distance metric can be of two types: $ {} ^ {10} S _ {3} ^ {1} $ or $ {} ^ {11} S _ {3} ^ {1} $, differing by their metrics in bundles of planes: in the first, a Euclidean, in the second, a pseudo-Euclidean metric of index 1. The metrics on the $ 2 $-planes coincide; it is a pseudo-Euclidean metric of index 1.

The quasi-hyperbolic $ 3 $-space $ {} ^ {11} S _ {3} ^ {1} $ can be interpreted as the group of motions of the pseudo-Euclidean $ 2 $-plane of index 1. The manifold of hyperbolic lines on this quasi-hyperbolic $ 3 $-space can be interpreted as a pair of such pseudo-Euclidean planes. The spaces $ {} ^ {10} S _ {3} ^ {1} $ and $ {} ^ {01} S _ {3} ^ {1} $ are dual to each other, and can be interpreted on a complex $ 2 $-plane.

References

[1]	B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)
[2]	I.M. Yaglom, B.A. Rozenfel'd, E.U. Yasinskaya, "Projective metrics" Russian Math. Surveys , 19 : 5 (1964) pp. 49–107 Uspekhi Mat. Nauk , 19 : 5 (1964) pp. 51–113
[a1]	O. Giering, "Vorlesungen über höhere Geometrie" , Vieweg (1982)
[a2]	B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)