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Difference between revisions of "Curvature tensor"

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A tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027360/c0273601.png" /> obtained by decomposing the [[Curvature form|curvature form]] in a local co-basis on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027360/c0273602.png" />. In particular, in a holonomic co-basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027360/c0273603.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027360/c0273604.png" />, the components of the curvature tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027360/c0273605.png" /> of an affine connection are expressed in terms of the Christoffel symbols of the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027360/c0273606.png" /> and their derivatives:
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A tensor of type $(1,3)$ obtained by decomposing the [[Curvature form|curvature form]] in a local co-basis on a manifold $M^n$. In particular, in a holonomic co-basis $dx^i$, $i=1,\dots,n$, the components of the curvature tensor $R_{lij}^k$ of an affine connection are expressed in terms of the Christoffel symbols of the connection $\Gamma_{ij}^k$ and their derivatives:
  
<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/c/c027/c027360/c0273607.png" /></td> </tr></table>
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$$R_{lij}^k=\partial_i\Gamma_{jl}^k-\partial_j\Gamma_{il}^k+\Gamma_{ip}^k\Gamma_{jl}^p-\Gamma_{jp}^k\Gamma_{il}^p.$$
  
In similar fashion one defines the curvature tensor for an arbitrary connection on a principal fibre space with structure Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027360/c0273608.png" /> in terms of a decomposition of the appropriate curvature form; this applies, in particular, to conformal and projective connections. It takes values in the Lie algebra of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027360/c0273609.png" /> and is an example of a so-called tensor with non-scalar components.
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In similar fashion one defines the curvature tensor for an arbitrary connection on a principal fibre space with structure Lie group $G$ in terms of a decomposition of the appropriate curvature form; this applies, in particular, to conformal and projective connections. It takes values in the Lie algebra of the group $G$ and is an example of a so-called tensor with non-scalar components.
  
 
For references see [[Curvature|Curvature]].
 
For references see [[Curvature|Curvature]].

Latest revision as of 10:01, 20 September 2014

A tensor of type $(1,3)$ obtained by decomposing the curvature form in a local co-basis on a manifold $M^n$. In particular, in a holonomic co-basis $dx^i$, $i=1,\dots,n$, the components of the curvature tensor $R_{lij}^k$ of an affine connection are expressed in terms of the Christoffel symbols of the connection $\Gamma_{ij}^k$ and their derivatives:

$$R_{lij}^k=\partial_i\Gamma_{jl}^k-\partial_j\Gamma_{il}^k+\Gamma_{ip}^k\Gamma_{jl}^p-\Gamma_{jp}^k\Gamma_{il}^p.$$

In similar fashion one defines the curvature tensor for an arbitrary connection on a principal fibre space with structure Lie group $G$ in terms of a decomposition of the appropriate curvature form; this applies, in particular, to conformal and projective connections. It takes values in the Lie algebra of the group $G$ and is an example of a so-called tensor with non-scalar components.

For references see Curvature.

How to Cite This Entry:
Curvature tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curvature_tensor&oldid=33345
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article