Flag space

A projective $ n $- space whose metric is defined by an absolute consisting of a collection of $ m $- planes, $ m = 0 \dots n - 1 $, imbedded in one another, called a flag; a flag space is denoted by $ F _ {n} $. The absolute of a flag space can be obtained from the absolutes of Galilean or pseudo-Galilean spaces by means of a passing to the limit in the quadrics of the absolutes. In particular, the flag (absolute) of the space $ F _ {3} $ consists of a $ 2 $- plane $ T _ {0} $, in which lies a line $ T _ {1} $( a Euclidean line), and on the line a point $ T _ {2} $. The plane $ F _ {2} $ is a projective $ 2 $- plane with a distinguished line $ T _ {0} $ and a distinguished point $ T _ {1} $ and coincides with Yaglom's Galilean plane. $ F _ {1} $ is a projective line with a distinguished point $ T _ {0} $, and is the same as the Euclidean line.

If one chooses an affine coordinate system $ ( x ^ {i} ) $ in a flag space $ F _ {n} $ so that the vectors of the lines passing through the $ ( n - m - 1) $- plane $ T _ {m} $ are defined by the condition $ x ^ {1} = {} \dots = x ^ {m} = 0 $, then one takes the number $ d = | x ^ {1} - y ^ {1} | $ as the distance between the points $ ( x ^ {1} \dots x ^ {n} ) $ and $ ( y ^ {1} \dots y ^ {n} ) $; if $ y ^ {1} = x ^ {1} \dots y ^ {k - 1 } = x ^ {k - 1 } $, then the distance is defined by the number $ d ^ {( k - 1) } = | x ^ {k} - y ^ {k} | $.

Lines that intersect the $ ( n - m) $- plane and not the $ ( n - m - 1) $- plane are called lines of order $ m $.

Motions of a flag space are collineations mapping the absolute into itself. The motions of a flag space form a subgroup of the affine transformations of an affine $ n $- space, and this group of motions of a flag space is a Lie group.

A space $ F _ {n} $ is self-dual. As the value of the angle between two $ ( n - 1) $- planes one takes the distance between the points dual to these planes.

A flag space is a special case of a semi-elliptic space. In particular, the flag space $ F _ {3} $ is the same as the $ 3 $- space $ S _ {3} ^ {012} $. The flag $ 3 $- space is the unique space with parabolic distance metrics on lines, in semi-planes and in bundles of planes.

References

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

Comments

$ F _ {2} $, or Yaglom's Galilean plane, may be described as follows. Recall Poncelet's description of the Euclidean plane as a projective plane with a metric determined by two "circular points" $ I $ and $ J $ whose join $ o $ is the line at infinity. I.M. Yaglom considered a modification in which $ I = J $ while $ o $ is still a special line through $ J $. In other words, the new metric is determined by the flag $ J o $. The role of circles, which were conics (cf. Cone) through $ I $ and $ J $, is taken over by conics touching $ o $ at $ J $, i.e. by parabolas whose diameters all have the same direction (cf. Parabola). Lines in that direction (through $ J $) behave differently from other lines. In general, $ F _ {n} $ contains lines of $ n $ distinct types.

For absolute of a space see Absolute; see also Galilean space; Pseudo-Galilean space.

References

[a1]	H.S.M. Coxeter, "The affine aspect of Yaglom's Galilean Feuerbach" Nieuw Archief voor Wiskunde (4) , 1 (1983) pp. 212–223
[a2]	I.M. Yaglom, "A simple non-Euclidean geometry and its physical basis" , Springer (1979) (Translated from Russian)