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Difference between revisions of "Semi-Euclidean space"

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A real affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s0840901.png" />-space equipped with a scalar product of vectors such that, relative to a suitably chosen basis, the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s0840902.png" /> of any vector with itself has the form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s0840903.png" /></td> </tr></table>
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Under these conditions, the semi-Euclidean space is said to have index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s0840904.png" /> and deficiency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s0840905.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s0840906.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s0840907.png" />, the expression for the scalar product of a vector with itself is a semi-definite quadratic form and the space is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s08409010.png" />-space of deficiency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s08409011.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s08409012.png" />.
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A real affine  $  n $-
 +
space equipped with a scalar product of vectors such that, relative to a suitably chosen basis, the scalar product  $  ( \mathbf x , \mathbf x ) $
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of any vector with itself has the form
 +
 
 +
$$
 +
( \mathbf x , \mathbf x )  =  - \sum _ { i= } 1 ^ { l }  ( x  ^ {i} )  ^ {2} + \sum _ { j= } l+ 1 ^ { n- }  d ( x  ^ {j} )  ^ {2} .
 +
$$
 +
 
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Under these conditions, the semi-Euclidean space is said to have index $  l $
 +
and deficiency $  d $
 +
and is denoted by $  {} ^ {l + ( d ) } R _ {n} $.  
 +
If $  l = 0 $,  
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the expression for the scalar product of a vector with itself is a semi-definite quadratic form and the space is called an $  n $-
 +
space of deficiency $  d $,  
 +
denoted by $  {} ^ {( d ) } R _ {n} $.
  
 
In the projective classification, a semi-Euclidean space can be defined as a [[Semi-elliptic space|semi-elliptic space]] or a [[Semi-hyperbolic space|semi-hyperbolic space]] with an improper absolute plane; these are spaces with projective metrics of the most general form.
 
In the projective classification, a semi-Euclidean space can be defined as a [[Semi-elliptic space|semi-elliptic space]] or a [[Semi-hyperbolic space|semi-hyperbolic space]] with an improper absolute plane; these are spaces with projective metrics of the most general form.
  
One defines a semi-non-Euclidean space as a metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s08409013.png" />-space which is a hypersphere with identified antipodal points in the semi-Euclidean space of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s08409014.png" /> and deficiency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s08409015.png" />. Thus, semi-elliptic and semi-hyperbolic spaces may be interpreted as hyperspheres of the above type (i.e. as semi-non-Euclidean spaces) in semi-Euclidean spaces of appropriate index and deficiency.
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One defines a semi-non-Euclidean space as a metric $  n $-
 +
space which is a hypersphere with identified antipodal points in the semi-Euclidean space of index $  l $
 +
and deficiency $  d $.  
 +
Thus, semi-elliptic and semi-hyperbolic spaces may be interpreted as hyperspheres of the above type (i.e. as semi-non-Euclidean spaces) in semi-Euclidean spaces of appropriate index and deficiency.
  
The geometrical interpretation of Galileo–Newton mechanics leads to the semi-Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084090/s08409016.png" /> (see [[#References|[2]]]).
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The geometrical interpretation of Galileo–Newton mechanics leads to the semi-Euclidean space $  {} ^ {( 1 ) } R _ {n} $(
 +
see [[#References|[2]]]).
  
 
A semi-Euclidean space is a [[Semi-Riemannian space|semi-Riemannian space]] of curvature zero.
 
A semi-Euclidean space is a [[Semi-Riemannian space|semi-Riemannian space]] of curvature zero.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.M.Y. Sommerville,  "Classification of geometries with projective metric"  ''Proc. Edinburgh Math. Soc.'' , '''28'''  (1910)  pp. 25–41</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.P. Kotel'nikov,  "The principle of relativity and Lobachevskii geometry" , ''In memoriam N.I. Lobachevskii'' , '''2''' , Kazan'  (1926)  (In Russian)</TD></TR><TR><TD valign="top">[3]</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">  D.M.Y. Sommerville,  "Classification of geometries with projective metric"  ''Proc. Edinburgh Math. Soc.'' , '''28'''  (1910)  pp. 25–41</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.P. Kotel'nikov,  "The principle of relativity and Lobachevskii geometry" , ''In memoriam N.I. Lobachevskii'' , '''2''' , Kazan'  (1926)  (In Russian)</TD></TR><TR><TD valign="top">[3]</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 real affine $ n $- space equipped with a scalar product of vectors such that, relative to a suitably chosen basis, the scalar product $ ( \mathbf x , \mathbf x ) $ of any vector with itself has the form

$$ ( \mathbf x , \mathbf x ) = - \sum _ { i= } 1 ^ { l } ( x ^ {i} ) ^ {2} + \sum _ { j= } l+ 1 ^ { n- } d ( x ^ {j} ) ^ {2} . $$

Under these conditions, the semi-Euclidean space is said to have index $ l $ and deficiency $ d $ and is denoted by $ {} ^ {l + ( d ) } R _ {n} $. If $ l = 0 $, the expression for the scalar product of a vector with itself is a semi-definite quadratic form and the space is called an $ n $- space of deficiency $ d $, denoted by $ {} ^ {( d ) } R _ {n} $.

In the projective classification, a semi-Euclidean space can be defined as a semi-elliptic space or a semi-hyperbolic space with an improper absolute plane; these are spaces with projective metrics of the most general form.

One defines a semi-non-Euclidean space as a metric $ n $- space which is a hypersphere with identified antipodal points in the semi-Euclidean space of index $ l $ and deficiency $ d $. Thus, semi-elliptic and semi-hyperbolic spaces may be interpreted as hyperspheres of the above type (i.e. as semi-non-Euclidean spaces) in semi-Euclidean spaces of appropriate index and deficiency.

The geometrical interpretation of Galileo–Newton mechanics leads to the semi-Euclidean space $ {} ^ {( 1 ) } R _ {n} $( see [2]).

A semi-Euclidean space is a semi-Riemannian space of curvature zero.

References

[1] D.M.Y. Sommerville, "Classification of geometries with projective metric" Proc. Edinburgh Math. Soc. , 28 (1910) pp. 25–41
[2] A.P. Kotel'nikov, "The principle of relativity and Lobachevskii geometry" , In memoriam N.I. Lobachevskii , 2 , Kazan' (1926) (In Russian)
[3] 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-Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-Euclidean_space&oldid=48651
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article