# 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)