Flag space

A projective -space whose metric is defined by an absolute consisting of a collection of -planes, , imbedded in one another, called a flag; a flag space is denoted by . 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 consists of a -plane , in which lies a line (a Euclidean line), and on the line a point . The plane is a projective -plane with a distinguished line and a distinguished point and coincides with Yaglom's Galilean plane. is a projective line with a distinguished point , and is the same as the Euclidean line.

If one chooses an affine coordinate system in a flag space so that the vectors of the lines passing through the -plane are defined by the condition , then one takes the number as the distance between the points and ; if , then the distance is defined by the number .

Lines that intersect the -plane and not the -plane are called lines of order .

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 -space, and this group of motions of a flag space is a Lie group.

A space is self-dual. As the value of the angle between two -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 is the same as the -space . The flag -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

, 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" and whose join is the line at infinity. I.M. Yaglom considered a modification in which while is still a special line through . In other words, the new metric is determined by the flag . The role of circles, which were conics (cf. Cone) through and , is taken over by conics touching at , i.e. by parabolas whose diameters all have the same direction (cf. Parabola). Lines in that direction (through ) behave differently from other lines. In general, contains lines of 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)