View source for Semi-pseudo-Riemannian space

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A manifold with a degenerate indefinite metric. The semi-pseudo-Riemannian manifold  $  {} ^ {l _ {1} \dots l _ {r} } V _ {n} ^ {m _ {1} \dots m _ {r - 1 }  } $
is defined as an  $  n $-
dimensional manifold with coordinates  $  x  ^ {i} $
in which there are given  $  r $
line elements

$$ 
ds _ {a}  ^ {2}  = \ 
\sum _ { i,j= } 1 ^ { {m _ a} - m _ {a-} 1 } g _ {( a) ij }  \ 
dx ^ {i + m _ {a-} 1 }  dx ^ {j + m _ {a-} 1 } ,
$$

where  $  0 = m _ {0} < m _ {1} < \dots < m _ {r} = n $; 
$  a = 1 \dots r $; 
and where the index of the quadratic form  $  g _ {( a) ij }  $
is  $  l _ {a} $. 
The line element  $  ds _ {a}  ^ {2} $
is defined for those vectors for which all components with indices smaller than  $  m _ {a - 1 }  + 1 $
or larger than  $  m _ {a} $
vanish. If  $  l _ {1} = l _ {2} = \dots = 0 $, 
a semi-pseudo-Riemannian space is a [[Semi-Riemannian space|semi-Riemannian space]]. The spaces  $  V _ {n}  ^ {m} $
and  $  {}  ^ {kl} {V _ {n}  ^ {m} } $
are quasi-Riemannian spaces. The basic concepts of differential geometry (for example, curvature) are defined in semi-pseudo-Riemannian spaces similarly to Riemannian spaces (see [[#References|[1]]]).

====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>