View source for Semi-Euclidean space

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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 ) $
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 [[#References|[2]]]).

A semi-Euclidean space is a [[Semi-Riemannian space|semi-Riemannian space]] of curvature zero.

====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>
<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>