Namespaces
Variants
Actions

Difference between revisions of "Semi-pseudo-Riemannian space"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (fixing subscripts)
Line 11: Line 11:
 
{{TEX|done}}
 
{{TEX|done}}
  
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 }  } $
+
A manifold with a degenerate indefinite metric. The semi-pseudo-Riemannian manifold  $  {} ^ {l _ {1} \cdots l _ {r} } V _ {n} ^ {m _ {1} \cdots m _ {r - 1 }  } $
is defined as an  $  n $-
+
is defined as an  $  n $-dimensional manifold with coordinates  $  x  ^ {i} $
dimensional manifold with coordinates  $  x  ^ {i} $
 
 
in which there are given  $  r $
 
in which there are given  $  r $
 
line elements
 
line elements
Line 19: Line 18:
 
$$  
 
$$  
 
ds _ {a}  ^ {2}  = \  
 
ds _ {a}  ^ {2}  = \  
\sum _ { i,j= } 1 ^ { {m _ a} - m _ {a-} 1 } g _ {( a) ij }  \  
+
\sum _ { i,j= } 1 ^ { {m _ a} - m _ {a- 1} } g _ {( a) ij }  \  
dx ^ {i + m _ {a-} 1 }  dx ^ {j + m _ {a-} 1 } ,
+
dx ^ {i + m _ {a- 1} }  dx ^ {j + m _ {a- 1} } ,
 
$$
 
$$
  

Revision as of 23:44, 7 May 2022


A manifold with a degenerate indefinite metric. The semi-pseudo-Riemannian manifold $ {} ^ {l _ {1} \cdots l _ {r} } V _ {n} ^ {m _ {1} \cdots 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. 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 [1]).

References

[1] 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-pseudo-Riemannian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-pseudo-Riemannian_space&oldid=48665
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article