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Difference between revisions of "Scalar curvature"

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''of a Riemannian manifold at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083250/s0832501.png" />''
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The trace of the [[Ricci tensor|Ricci tensor]] with respect to the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083250/s0832502.png" />. The scalar curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083250/s0832503.png" /> is connected with the [[Ricci curvature|Ricci curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083250/s0832504.png" /> and the [[Sectional curvature|sectional curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083250/s0832505.png" /> by the formulas
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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/s083/s083250/s0832506.png" /></td> </tr></table>
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''of a Riemannian manifold at a point  $  p $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083250/s0832507.png" /> is an orthonormal basis of the tangent space. In the equivalent Einstein notation, these equations have the form
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The trace of the [[Ricci tensor|Ricci tensor]] with respect to the metric tensor  $  g $.  
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The scalar curvature  $  s ( p) $
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is connected with the [[Ricci curvature|Ricci curvature]]  $  r $
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and the [[Sectional curvature|sectional curvature]]  $  k $
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by the formulas
  
<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/s083/s083250/s0832508.png" /></td> </tr></table>
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$$
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s ( p)  = \
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\sum _ {i = 1 } ^ { n }  r ( e _ {i} )  = \
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\sum _ {i, j = 1 } ^ { n }  k ( e _ {i} , e _ {j} ),
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083250/s0832509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083250/s08325010.png" /> are the components of the Ricci tensor and the curvature tensor, respectively, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083250/s08325011.png" /> are the contravariant components of the metric tensor.
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where $  e _ {1} \dots e _ {n} $
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is an orthonormal basis of the tangent space. In the equivalent Einstein notation, these equations have the form
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$$
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s ( p)  = g  ^ {ij} R _ {ij}  = \
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g  ^ {ij} g  ^ {kl} R _ {kijl} ,
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$$
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where  $  R _ {ij} $
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and  $  R _ {kijl} $
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are the components of the Ricci tensor and the curvature tensor, respectively, and the $  g  ^ {ij} $
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are the contravariant components of the metric tensor.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1963–1969)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1963–1969)</TD></TR></table>

Latest revision as of 08:12, 6 June 2020


of a Riemannian manifold at a point $ p $

The trace of the Ricci tensor with respect to the metric tensor $ g $. The scalar curvature $ s ( p) $ is connected with the Ricci curvature $ r $ and the sectional curvature $ k $ by the formulas

$$ s ( p) = \ \sum _ {i = 1 } ^ { n } r ( e _ {i} ) = \ \sum _ {i, j = 1 } ^ { n } k ( e _ {i} , e _ {j} ), $$

where $ e _ {1} \dots e _ {n} $ is an orthonormal basis of the tangent space. In the equivalent Einstein notation, these equations have the form

$$ s ( p) = g ^ {ij} R _ {ij} = \ g ^ {ij} g ^ {kl} R _ {kijl} , $$

where $ R _ {ij} $ and $ R _ {kijl} $ are the components of the Ricci tensor and the curvature tensor, respectively, and the $ g ^ {ij} $ are the contravariant components of the metric tensor.

References

[1] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[2] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)

Comments

References

[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963–1969)
How to Cite This Entry:
Scalar curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Scalar_curvature&oldid=12835
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article