Namespaces
Variants
Actions

Difference between revisions of "Torsion form"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(Tex done)
 
Line 1: Line 1:
The covariant differential of the vector-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093310/t0933101.png" />-form of the displacement of an [[Affine connection|affine connection]], the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093310/t0933102.png" />-form
+
The [[covariant differential]] of the vector-valued $1$-form of the displacement of an [[affine connection]], the $2$-form
 
+
$$
<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/t093310/t0933103.png" /></td> </tr></table>
+
\Omega = D \omega = d \omega + \theta \wedge \omega
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093310/t0933104.png" /> is the connection form. The torsion form satisfies the first Bianchi identity:
+
where $\theta$ is the connection form. The torsion form satisfies the first [[Bianchi identity]]:
 
+
$$
<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/t093310/t0933105.png" /></td> </tr></table>
+
d \Omega = \theta \wedge \Omega + \omega \wedge \Theta
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093310/t0933106.png" /> is the [[Curvature form|curvature form]] of the given connection. The definition of a torsion form for reductive connections is analogous.
+
where $\Theta$ is the [[curvature form]] of the given connection. The definition of a torsion form for reductive connections is analogous.
  
  
Line 15: Line 15:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1963)</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)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 19:51, 6 October 2016

The covariant differential of the vector-valued $1$-form of the displacement of an affine connection, the $2$-form $$ \Omega = D \omega = d \omega + \theta \wedge \omega $$ where $\theta$ is the connection form. The torsion form satisfies the first Bianchi identity: $$ d \Omega = \theta \wedge \Omega + \omega \wedge \Theta $$ where $\Theta$ is the curvature form of the given connection. The definition of a torsion form for reductive connections is analogous.


Comments

References

[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963)
How to Cite This Entry:
Torsion form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion_form&oldid=39365
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article