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A projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p0756901.png" />-space (cf. [[Projective space|Projective space]]) with a distinguished infinitely-distant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p0756902.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p0756903.png" /> in the affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p0756904.png" />-space (cf. [[Affine space|Affine space]]) in which in turn an infinitely-distant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p0756905.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p0756906.png" /> of the [[Pseudo-Euclidean space|pseudo-Euclidean space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p0756907.png" /> has been distinguished, while in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p0756908.png" /> an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p0756909.png" />-quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569010.png" /> has been distinguished which is the absolute of the hyperbolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569011.png" />-space of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569012.png" />. The family of planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569013.png" /> and quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569014.png" /> forms the [[Absolute|absolute]] (basis) of the pseudo-Galilean space; the latter is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569015.png" />. E.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569016.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569017.png" /> has as absolute a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569018.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569019.png" />, a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569021.png" /> and a pair of real points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569023.png" />. A pseudo-Galilean space can be defined as an affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569024.png" />-space in whose infinitely-distant hyperbolic hyperplane under completion to projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569025.png" />-space the geometry of the pseudo-Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569026.png" />-space of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569027.png" /> has been defined.
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A projective  $  n $-
 +
space (cf. [[Projective space|Projective space]]) with a distinguished infinitely-distant $  ( n - 1 ) $-
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plane $  T _ {0} $
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in the affine $  n $-
 +
space (cf. [[Affine space|Affine space]]) in which in turn an infinitely-distant $  ( n - 2 ) $-
 +
plane $  T _ {1} $
 +
of the [[Pseudo-Euclidean space|pseudo-Euclidean space]] $  {}  ^ {l} R _ {n-} 1 $
 +
has been distinguished, while in $  T _ {1} $
 +
an $  ( n - 3 ) $-
 +
quadric $  Q _ {2} $
 +
has been distinguished which is the absolute of the hyperbolic $  ( n - 1 ) $-
 +
space of index $  l $.  
 +
The family of planes $  T _ {0} , T _ {1} $
 +
and quadric $  Q _ {2} $
 +
forms the [[Absolute|absolute]] (basis) of the pseudo-Galilean space; the latter is denoted by $  {}  ^ {l} \Gamma _ {n} $.  
 +
E.g., $  3 $-
 +
space $  {}  ^ {1} \Gamma _ {3} $
 +
has as absolute a $  2 $-
 +
plane $  T _ {0} $,  
 +
a straight line $  T _ {1} $
 +
in $  T _ {0} $
 +
and a pair of real points $  Q _ {2} $
 +
on $  T _ {1} $.  
 +
A pseudo-Galilean space can be defined as an affine $  n $-
 +
space in whose infinitely-distant hyperbolic hyperplane under completion to projective $  n $-
 +
space the geometry of the pseudo-Euclidean $  ( n - 1 ) $-
 +
space of index $  l $
 +
has been defined.
  
 
The distance between points is defined analogously to the distance in a [[Galilean space|Galilean space]].
 
The distance between points is defined analogously to the distance in a [[Galilean space|Galilean space]].
  
The motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569028.png" /> are its collineations mapping the absolute into itself. The motions form a group, which is a Lie group.
+
The motions of $  {}  ^ {l} \Gamma _ {n} $
 +
are its collineations mapping the absolute into itself. The motions form a group, which is a Lie group.
  
The space whose absolute is dual to the absolute of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569029.png" /> is called a [[Co-pseudo-Galilean space|co-pseudo-Galilean space]]. A [[Flag space|flag space]] is a limit case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075690/p07569030.png" />.
+
The space whose absolute is dual to the absolute of $  {}  ^ {l} \Gamma _ {n} $
 +
is called a [[Co-pseudo-Galilean space|co-pseudo-Galilean space]]. A [[Flag space|flag space]] is a limit case of $  {}  ^ {l} \Gamma _ {n} $.
  
 
====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====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Non-Euclidean geometry" , Univ. Toronto Press  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)  pp. Chapt. X</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Non-Euclidean geometry" , Univ. Toronto Press  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)  pp. Chapt. X</TD></TR></table>

Revision as of 08:08, 6 June 2020


A projective $ n $- space (cf. Projective space) with a distinguished infinitely-distant $ ( n - 1 ) $- plane $ T _ {0} $ in the affine $ n $- space (cf. Affine space) in which in turn an infinitely-distant $ ( n - 2 ) $- plane $ T _ {1} $ of the pseudo-Euclidean space $ {} ^ {l} R _ {n-} 1 $ has been distinguished, while in $ T _ {1} $ an $ ( n - 3 ) $- quadric $ Q _ {2} $ has been distinguished which is the absolute of the hyperbolic $ ( n - 1 ) $- space of index $ l $. The family of planes $ T _ {0} , T _ {1} $ and quadric $ Q _ {2} $ forms the absolute (basis) of the pseudo-Galilean space; the latter is denoted by $ {} ^ {l} \Gamma _ {n} $. E.g., $ 3 $- space $ {} ^ {1} \Gamma _ {3} $ has as absolute a $ 2 $- plane $ T _ {0} $, a straight line $ T _ {1} $ in $ T _ {0} $ and a pair of real points $ Q _ {2} $ on $ T _ {1} $. A pseudo-Galilean space can be defined as an affine $ n $- space in whose infinitely-distant hyperbolic hyperplane under completion to projective $ n $- space the geometry of the pseudo-Euclidean $ ( n - 1 ) $- space of index $ l $ has been defined.

The distance between points is defined analogously to the distance in a Galilean space.

The motions of $ {} ^ {l} \Gamma _ {n} $ are its collineations mapping the absolute into itself. The motions form a group, which is a Lie group.

The space whose absolute is dual to the absolute of $ {} ^ {l} \Gamma _ {n} $ is called a co-pseudo-Galilean space. A flag space is a limit case of $ {} ^ {l} \Gamma _ {n} $.

References

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

Comments

References

[a1] H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1968)
[a2] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) pp. Chapt. X
How to Cite This Entry:
Pseudo-Galilean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-Galilean_space&oldid=48341
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article