# 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) |

#### 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 |

**How to Cite This Entry:**

Pseudo-Galilean space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Pseudo-Galilean_space&oldid=48341