Semi-hyperbolic space
A projective -space in which the metric is defined by a given absolute consisting of the following collection: a second-order real cone
of index
with an
-plane vertex
; a real
-cone
of index
with an
-plane vertex
in the
-plane
;
; a real
-cone
of index
with an
-plane vertex
; and a non-degenerate real
-quadric
of index
in the plane
;
. This is the definition of a semi-hyperbolic space with indices
; it is denoted by
.
If the cone is a pair of merging planes, both identical with
(for
), the semi-hyperbolic plane with the improper plane
is called a semi-Euclidean space:
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The distance between two points and
is defined as a function of the position of the straight line
relative to the planes
. In particular, if
does not intersect
, the distance between
and
is defined through a scalar product, in analogy with the appropriate definition in a quasi-hyperbolic space. If
intersects
but does not intersect
, or it intersects
but does not intersect
, the distance between the points is defined as the scalar product with itself of the distance between the vectors of the points
and
.
Depending on the position of the absolute relative to the planes 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 -plane
is identical with the pseudo-Euclidean space
, the plane
— with the co-pseudo-Euclidean space
; the
-spaces
and
coincide with the quasi-hyperbolic
-space, the
-space
— with the co-pseudo-Euclidean space
, etc. The
-space
is dual to the pseudo-Galilean space
, it is known as a co-pseudo-Galilean space; its absolute consists of pairs of real planes (a cone
) and a point
on the straight line
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 and
, 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) |
Comments
References
[a1] | B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian) |
Semi-hyperbolic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-hyperbolic_space&oldid=48661