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A manifold with a degenerate indefinite metric. The semi-pseudo-Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s0842901.png" /> is defined as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s0842902.png" />-dimensional manifold with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s0842903.png" /> in which there are given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s0842904.png" /> line elements
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s0842905.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s0842906.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s0842907.png" />; and where the index of the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s0842908.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s0842909.png" />. The line element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s08429010.png" /> is defined for those vectors for which all components with indices smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s08429011.png" /> or larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s08429012.png" /> vanish. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s08429013.png" />, a semi-pseudo-Riemannian space is a [[Semi-Riemannian space|semi-Riemannian space]]. The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s08429014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084290/s08429015.png" /> 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]]]).
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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 }  } $
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is defined as an  $  n $-
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dimensional manifold with coordinates  $  x  ^ {i} $
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in which there are given  $  r $
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line elements
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$$
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ds _ {a}  ^ {2}  = \
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\sum _ { i,j= } 1 ^ { {m _ a} - m _ {a-} 1 } g _ {( a) ij }  \
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dx ^ {i + m _ {a-} 1 }  dx ^ {j + m _ {a-} 1 } ,
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$$
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where  $  0 = m _ {0} < m _ {1} < \dots < m _ {r} = n $;
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$  a = 1 \dots r $;  
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and where the index of the quadratic form $  g _ {( a) ij }  $
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is $  l _ {a} $.  
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The line element $  ds _ {a}  ^ {2} $
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is defined for those vectors for which all components with indices smaller than $  m _ {a - 1 }  + 1 $
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or larger than $  m _ {a} $
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vanish. If $  l _ {1} = l _ {2} = \dots = 0 $,  
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a semi-pseudo-Riemannian space is a [[Semi-Riemannian space|semi-Riemannian space]]. The spaces $  V _ {n}  ^ {m} $
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and $  {}  ^ {kl} {V _ {n}  ^ {m} } $
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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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Revision as of 08:13, 6 June 2020


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. 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