Pseudo-Galilean space

A projective $n$- space (cf. Projective space) with a distinguished infinitely-distant $( n - 1 )$- plane $T _ {0}$ in the affine $n$- space (cf. Affine space) in which in turn an infinitely-distant $( n - 2 )$- plane $T _ {1}$ of the pseudo-Euclidean space ${} ^ {l} R _ {n-} 1$ has been distinguished, while in $T _ {1}$ an $( n - 3 )$- quadric $Q _ {2}$ has been distinguished which is the absolute of the hyperbolic $( n - 1 )$- space of index $l$. The family of planes $T _ {0} , T _ {1}$ and quadric $Q _ {2}$ forms the absolute (basis) of the pseudo-Galilean space; the latter is denoted by ${} ^ {l} \Gamma _ {n}$. E.g., $3$- space ${} ^ {1} \Gamma _ {3}$ has as absolute a $2$- plane $T _ {0}$, a straight line $T _ {1}$ in $T _ {0}$ and a pair of real points $Q _ {2}$ on $T _ {1}$. A pseudo-Galilean space can be defined as an affine $n$- space in whose infinitely-distant hyperbolic hyperplane under completion to projective $n$- space the geometry of the pseudo-Euclidean $( n - 1 )$- space of index $l$ has been defined.

The distance between points is defined analogously to the distance in a Galilean space.

The motions of ${} ^ {l} \Gamma _ {n}$ 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 ${} ^ {l} \Gamma _ {n}$ is called a co-pseudo-Galilean space. A flag space is a limit case of ${} ^ {l} \Gamma _ {n}$.

References

[1]	B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)
[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