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