Namespaces
Variants
Actions

Difference between revisions of "Flag space"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f0405602.png" />-space whose metric is defined by an absolute consisting of a collection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f0405603.png" />-planes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f0405604.png" />, imbedded in one another, called a flag; a flag space is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f0405605.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f0405606.png" /> consists of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f0405607.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f0405608.png" />, in which lies a line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f0405609.png" /> (a Euclidean line), and on the line a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056010.png" />. The plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056011.png" /> is a projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056012.png" />-plane with a distinguished line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056013.png" /> and a distinguished point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056014.png" /> and coincides with Yaglom's Galilean plane. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056015.png" /> is a projective line with a distinguished point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056016.png" />, and is the same as the Euclidean line.
+
<!--
 +
f0405602.png
 +
$#A+1 = 50 n = 0
 +
$#C+1 = 50 : ~/encyclopedia/old_files/data/F040/F.0400560 Flag space
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
If one chooses an affine coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056017.png" /> in a flag space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056018.png" /> so that the vectors of the lines passing through the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056019.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056020.png" /> are defined by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056021.png" />, then one takes the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056022.png" /> as the distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056024.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056025.png" />, then the distance is defined by the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056026.png" />.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
Lines that intersect the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056027.png" />-plane and not the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056028.png" />-plane are called lines of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056030.png" />.
+
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.
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056032.png" />-space, and this group of motions of a flag space is a Lie group.
+
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} | $.
  
A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056033.png" /> is self-dual. As the value of the angle between two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056034.png" />-planes one takes the distance between the points dual to these planes.
+
Lines that intersect the  $  ( n - m) $-
 +
plane and not the $  ( n - m - 1) $-
 +
plane are called lines of order  $  m $.
  
A flag space is a special case of a [[Semi-elliptic space|semi-elliptic space]]. In particular, the flag space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056035.png" /> is the same as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056036.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056037.png" />. The flag <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056038.png" />-space is the unique space with parabolic distance metrics on lines, in semi-planes and in bundles of planes.
+
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|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056039.png" />, 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"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056041.png" /> whose join <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056042.png" /> is the line at infinity. I.M. Yaglom considered a modification in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056043.png" /> while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056044.png" /> is still a special line through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056045.png" />. In other words, the new metric is determined by the [[Flag|flag]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056046.png" />. The role of circles, which were conics (cf. [[Cone|Cone]]) through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056048.png" />, is taken over by conics touching <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056049.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056050.png" />, i.e. by parabolas whose diameters all have the same direction (cf. [[Parabola|Parabola]]). Lines in that direction (through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056051.png" />) behave differently from other lines. In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056052.png" /> contains lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040560/f04056053.png" /> distinct types.
+
$  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|flag]] $  J o $.  
 +
The role of circles, which were conics (cf. [[Cone|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|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|Absolute]]; see also [[Galilean space|Galilean space]]; [[Pseudo-Galilean space|Pseudo-Galilean space]].
 
For absolute of a space see [[Absolute|Absolute]]; see also [[Galilean space|Galilean space]]; [[Pseudo-Galilean space|Pseudo-Galilean space]].

Latest revision as of 19:39, 5 June 2020


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