Semi-elliptic space
A projective -space in which the metric is defined by a given absolute, which is the aggregate of an imaginary quadratic cone
with an
-flat vertex
, an
-imaginary cone
with an
-flat vertex
in the
-plane
, etc., up to an
-imaginary cone
with an
-flat vertex
and a non-degenerate imaginary
-quadratic
in the
-plane
,
. The indices of the cones
,
, are:
,
;
. A semi-elliptic space is denoted by
.
In case the cone is a pair of merging planes coinciding with the plane
(for
), the space with the improper plane
is called the semi-Euclidean space
.
The distance between two points and
is defined according to the position of the straight line
with respect to the planes
. If, in particular, the line
does not intersect the plane
, then the distance between the points
and
is defined in terms of the scalar product, analogously to the distance in a quasi-elliptic space. If, however, the line
intersects the plane
but does not intersect the plane
, or intersects the plane
but does not intersect the plane
, the distance between the points is defined using the scalar square of the difference of the corresponding vectors of the points
and
.
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 -plane
coincides with the Euclidean and
with the co-Euclidean plane; the
-space
with the quasi-elliptic and
with the Euclidean
-space; the
-space
is Galilean,
is a flag space, etc. The
-space
corresponds by the duality principle to the Galilean
-space
and is called the co-Galilean space. (The absolute of a co-Galilean space consists of a pair of imaginary planes (the cone
) and a point
on the straight line
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 ,
, 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) |
Comments
References
[a1] | B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian) |
Semi-elliptic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-elliptic_space&oldid=32978