Pseudo-Galilean space
A projective -space (cf. Projective space) with a distinguished infinitely-distant
-plane
in the affine
-space (cf. Affine space) in which in turn an infinitely-distant
-plane
of the pseudo-Euclidean space
has been distinguished, while in
an
-quadric
has been distinguished which is the absolute of the hyperbolic
-space of index
. The family of planes
and quadric
forms the absolute (basis) of the pseudo-Galilean space; the latter is denoted by
. E.g.,
-space
has as absolute a
-plane
, a straight line
in
and a pair of real points
on
. A pseudo-Galilean space can be defined as an affine
-space in whose infinitely-distant hyperbolic hyperplane under completion to projective
-space the geometry of the pseudo-Euclidean
-space of index
has been defined.
The distance between points is defined analogously to the distance in a Galilean space.
The motions of are its collineations mapping the absolute into itself. The motions form a group, which is a Lie group.
The space whose absolute is dual to the absolute of is called a co-pseudo-Galilean space. A flag space is a limit case of
.
References
[1] | B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian) |
Comments
References
[a1] | H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1968) |
[a2] | S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) pp. Chapt. X |
Pseudo-Galilean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-Galilean_space&oldid=48341