Namespaces
Variants
Actions

Difference between revisions of "Semi-symplectic 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/s/s084/s084430/s0844301.png" />-space in which there is given a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s0844302.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s0844303.png" />, in this a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s0844304.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s0844305.png" />, etc., up to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s0844306.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s0844307.png" />, where in the space a null-system is given, taking all the points of the space to planes passing through the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s0844308.png" />; the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s0844309.png" /> is given an absolute null-system taking all its points to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s08443010.png" />-planes lying in it and passing through the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s08443011.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s08443012.png" />, etc., up to an absolute null-system of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s08443013.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s08443014.png" />, taking all its points to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s08443015.png" />-planes lying in it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s08443016.png" />. This semi-symplectic space is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s08443017.png" />.
+
<!--
 +
s0844301.png
 +
$#A+1 = 18 n = 0
 +
$#C+1 = 18 : ~/encyclopedia/old_files/data/S084/S.0804430 Semi\AAhsymplectic space
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
A projective  $  ( 2n + 1) $-
 +
space in which there is given a $  ( 2n - 2m _ {0} - 1) $-
 +
plane $  T _ {0} $,  
 +
in this a $  ( 2n - 2m _ {1} - 1) $-
 +
plane $  T _ {1} $,  
 +
etc., up to a $  ( 2n - 2m _ {r - 1 }  - 1) $-
 +
plane $  T _ {r - 1 }  $,  
 +
where in the space a null-system is given, taking all the points of the space to planes passing through the plane $  T _ {0} $;  
 +
the plane $  T _ {0} $
 +
is given an absolute null-system taking all its points to $  ( 2n - 2m _ {0} - 2) $-
 +
planes lying in it and passing through the $  ( 2n - 2m _ {1} - 1) $-
 +
plane $  T _ {1} $,  
 +
etc., up to an absolute null-system of the $  ( 2n - 2m _ {r - 1 }  - 1) $-
 +
plane $  T _ {r - 1 }  $,  
 +
taking all its points to $  ( 2n - 2m _ {r - 1 }  - 2) $-
 +
planes lying in it, $  0 \leq  m _ {0} < m _ {1} < \dots < m _ {r - 1 }  < n $.  
 +
This semi-symplectic space is denoted by $  \mathop{\rm Sp} _ {2n + 1 }  ^ {2m _ {0} + 1 \dots 2m _ {r - 1 }  + 1 } $.
  
 
A semi-symplectic space is obtained by a method analogous to the transition from elliptic and hyperbolic spaces to semi-elliptic and semi-hyperbolic spaces, and is more general than a quasi-symplectic space.
 
A semi-symplectic space is obtained by a method analogous to the transition from elliptic and hyperbolic spaces to semi-elliptic and semi-hyperbolic spaces, and is more general than a quasi-symplectic space.
  
The collineations of a semi-symplectic space that take the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084430/s08443018.png" /> to themselves and that commute with the null-systems are called semi-symplectic transformations of the semi-symplectic space.
+
The collineations of a semi-symplectic space that take the planes $  T _ {i} $
 +
to themselves and that commute with the null-systems are called semi-symplectic transformations of the semi-symplectic space.
  
 
There exist invariants of semi-symplectic transformations analogous to the symplectic invariants of symplectic spaces. The semi-symplectic transformations form a Lie group.
 
There exist invariants of semi-symplectic transformations analogous to the symplectic invariants of symplectic spaces. The semi-symplectic transformations form a Lie group.
Line 9: Line 38:
 
====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">  B.A. [B.A. Rozenfel'd] Rosenfel'd,  "A history of non-euclidean geometry" , Springer  (1988)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.A. [B.A. Rozenfel'd] Rosenfel'd,  "A history of non-euclidean geometry" , Springer  (1988)  (Translated from Russian)</TD></TR></table>

Revision as of 08:13, 6 June 2020


A projective $ ( 2n + 1) $- space in which there is given a $ ( 2n - 2m _ {0} - 1) $- plane $ T _ {0} $, in this a $ ( 2n - 2m _ {1} - 1) $- plane $ T _ {1} $, etc., up to a $ ( 2n - 2m _ {r - 1 } - 1) $- plane $ T _ {r - 1 } $, where in the space a null-system is given, taking all the points of the space to planes passing through the plane $ T _ {0} $; the plane $ T _ {0} $ is given an absolute null-system taking all its points to $ ( 2n - 2m _ {0} - 2) $- planes lying in it and passing through the $ ( 2n - 2m _ {1} - 1) $- plane $ T _ {1} $, etc., up to an absolute null-system of the $ ( 2n - 2m _ {r - 1 } - 1) $- plane $ T _ {r - 1 } $, taking all its points to $ ( 2n - 2m _ {r - 1 } - 2) $- planes lying in it, $ 0 \leq m _ {0} < m _ {1} < \dots < m _ {r - 1 } < n $. This semi-symplectic space is denoted by $ \mathop{\rm Sp} _ {2n + 1 } ^ {2m _ {0} + 1 \dots 2m _ {r - 1 } + 1 } $.

A semi-symplectic space is obtained by a method analogous to the transition from elliptic and hyperbolic spaces to semi-elliptic and semi-hyperbolic spaces, and is more general than a quasi-symplectic space.

The collineations of a semi-symplectic space that take the planes $ T _ {i} $ to themselves and that commute with the null-systems are called semi-symplectic transformations of the semi-symplectic space.

There exist invariants of semi-symplectic transformations analogous to the symplectic invariants of symplectic spaces. The semi-symplectic transformations form a Lie group.

References

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

Comments

References

[a1] B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)
How to Cite This Entry:
Semi-symplectic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-symplectic_space&oldid=13220
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article