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A tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093340/t0933401.png" /> that is skew-symmetric with respect to its indices, obtained by decomposing the [[Torsion form|torsion form]] of a connection in terms of a local cobasis on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093340/t0933402.png" />. In particular, in terms of a holonomic cobasis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093340/t0933403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093340/t0933404.png" />, the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093340/t0933405.png" /> of the torsion tensor are expressed in terms of the Christoffel symbols (cf. [[Christoffel symbol|Christoffel symbol]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093340/t0933406.png" /> of the connection as follows:
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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/t/t093/t093340/t0933407.png" /></td> </tr></table>
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A tensor of type  $  ( 1, 2) $
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that is skew-symmetric with respect to its indices, obtained by decomposing the [[Torsion form|torsion form]] of a connection in terms of a local cobasis on a manifold  $  M  ^ {n} $.
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In particular, in terms of a holonomic cobasis  $  dx  ^ {i} $,
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$  i = 1 \dots n $,
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the components  $  S _ {ij}  ^ {k} $
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of the torsion tensor are expressed in terms of the Christoffel symbols (cf. [[Christoffel symbol|Christoffel symbol]])  $  \Gamma _ {ij}  ^ {k} $
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of the connection as follows:
  
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$$
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S _ {ij}  ^ {k}  =  \Gamma _ {ij}  ^ {k} - \Gamma _ {ji}  ^ {k} .
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$$
  
 
====Comments====
 
====Comments====
In terms of covariant derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093340/t0933408.png" /> and vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093340/t0933409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093340/t09334010.png" /> the torsion tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093340/t09334011.png" /> can be described as follows:
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In terms of covariant derivatives $  \nabla $
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and vector fields $  X $,  
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$  Y $
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the torsion tensor $  T $
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can be described as follows:
  
<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/t/t093/t093340/t09334012.png" /></td> </tr></table>
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$$
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T ( X, Y)  = \nabla _ {X} Y - \nabla _ {Y} X - [ X, Y ] .
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$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.J. Hicks,  "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.J. Hicks,  "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR></table>

Latest revision as of 08:26, 6 June 2020


A tensor of type $ ( 1, 2) $ that is skew-symmetric with respect to its indices, obtained by decomposing the torsion form of a connection in terms of a local cobasis on a manifold $ M ^ {n} $. In particular, in terms of a holonomic cobasis $ dx ^ {i} $, $ i = 1 \dots n $, the components $ S _ {ij} ^ {k} $ of the torsion tensor are expressed in terms of the Christoffel symbols (cf. Christoffel symbol) $ \Gamma _ {ij} ^ {k} $ of the connection as follows:

$$ S _ {ij} ^ {k} = \Gamma _ {ij} ^ {k} - \Gamma _ {ji} ^ {k} . $$

Comments

In terms of covariant derivatives $ \nabla $ and vector fields $ X $, $ Y $ the torsion tensor $ T $ can be described as follows:

$$ T ( X, Y) = \nabla _ {X} Y - \nabla _ {Y} X - [ X, Y ] . $$

References

[a1] N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a2] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[a3] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
How to Cite This Entry:
Torsion tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion_tensor&oldid=49000
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article