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Difference between revisions of "Semi-Euclidean space"

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$$  
 
$$  
( \mathbf x , \mathbf x )  =  - \sum _ { i= } 1 ^ { l }  ( x  ^ {i} )  ^ {2} + \sum _ { j= } l+ 1 ^ { n- } d ( x  ^ {j} )  ^ {2} .
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( \mathbf x , \mathbf x )  =  - \sum _ {i=1}^ { l }  ( x  ^ {i} )  ^ {2} + \sum _ { j=l+ 1} ^ { n-d} ( x  ^ {j} )  ^ {2} .
 
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denoted by  $  {} ^ {( d ) } R _ {n} $.
 
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.
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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 $-
 
One defines a semi-non-Euclidean space as a metric  $  n $-
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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>
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<table>
 
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<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>
====Comments====
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<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>
 
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<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>
====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>
 

Latest revision as of 20:16, 10 January 2024


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)
[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=54967
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article