# Flag space

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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.

How to Cite This Entry:
Flag space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flag_space&oldid=14163
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article