# Flag space

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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)
How to Cite This Entry:
Flag space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flag_space&oldid=46939
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article