View source for Semi-pseudo-Euclidean space

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A vector space with a degenerate indefinite metric. The semi-pseudo-Euclidean space  $  {} ^ {l _ {1} \dots l _ {r} } R _ {n} ^ {m _ {1} \dots m _ {r - 1 }  } $
is defined as an  $  n $-
dimensional space in which there are given  $  r $
scalar products

$$ 
( x, y) _ {a}  = \ 
\sum \epsilon _ {i _ {a}  }
x ^ {i _ {a} } y ^ {i _ {a} } ,
$$

where  $  0 = m _ {0} < m _ {1} < \dots < m _ {r} = n $; 
$  a = 1 \dots r $; 
$  i _ {a} = m _ {a - 1 }  + 1 \dots m _ {a} $; 
$  \epsilon _ {i _ {a}  } = \pm  1 $, 
and  $  - 1 $
occurs  $  l _ {a} $
times among the numbers  $  \epsilon _ {i _ {a}  } $. 
The product  $  ( x, y) _ {a} $
is defined for those vectors for which all coordinates  $  x  ^ {i} $, 
$  i \leq  m _ {a - 1 }  $
or  $  i> m _ {a} + 1 $, 
are zero. The first scalar square of an arbitrary vector  $  x $
of a semi-pseudo-Euclidean space is a degenerate quadratic form in the vector coordinates:

$$ 
( x, x)  = \ 
- ( x _ {1} )  ^ {2} - \dots -
( x _ {l _ {1}  } )  ^ {2} +
( x _ {l _ {1}  + 1 } )  ^ {2} + \dots +
( x _ {n - d }  )  ^ {2} ,
$$

where  $  l $
is the index and  $  d = n - m _ {1} $
is the defect of the space. If  $  l _ {1} = \dots = l _ {r} = 0 $, 
the semi-pseudo-Euclidean space is a [[Semi-Euclidean space|semi-Euclidean space]]. Straight lines,  $  m $-
dimensional planes  $  ( m < n) $, 
parallelism, and length of vectors, are defined in semi-pseudo-Euclidean spaces in the same way as in pseudo-Euclidean spaces. In the semi-pseudo-Euclidean space  $  {} ^ {l + ( d) } {R _ {n} } $
one can choose an orthogonal basis consisting of  $  l $
vectors of imaginary length, of  $  n - l - d $
of real length and of  $  d $
isotropic vectors. Through every point of a semi-pseudo-Euclidean space of defect  $  d $
passes an  $  n $-
dimensional isotropic plane all vectors of which are orthogonal to all vectors of the space. See also [[Galilean space|Galilean space]].

====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>

====Comments====

====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>