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Difference between revisions of "Curvature form"

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A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027330/c0273301.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027330/c0273302.png" /> on a principal fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027330/c0273303.png" /> with structure Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027330/c0273304.png" />, taking values in the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027330/c0273305.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027330/c0273306.png" /> and defined by the [[Connection form|connection form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027330/c0273307.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027330/c0273308.png" /> by the formula
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A $2$-form $\Omega$ on a principal fibre bundle $P$ with structure Lie group $G$, taking values in the Lie algebra $\mathfrak g$ of the group $G$ and defined by the [[Connection form|connection form]] $\theta$ on $P$ by the formula
  
<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/c027330/c0273309.png" /></td> </tr></table>
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$$\Omega=d\theta+\frac12[\theta,\theta].$$
  
The curvature form is a measure of the deviation of the given connection from the locally flat connection characterized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027330/c02733010.png" />. It satisfies the Bianchi identity
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The curvature form is a measure of the deviation of the given connection from the locally flat connection characterized by the condition $\Omega\equiv0$. It satisfies the 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/c/c027/c027330/c02733011.png" /></td> </tr></table>
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$$d\Omega=[\Omega,\theta]$$
  
 
and defines the holonomy algebra (see [[Holonomy group|Holonomy group]]).
 
and defines the holonomy algebra (see [[Holonomy group|Holonomy group]]).
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====Comments====
 
====Comments====
The equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027330/c02733012.png" /> is called the structure equation.
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The equation $\Omega=d\theta+[\theta,\theta]/2$ is called the structure equation.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1''' , Interscience  (1963)  pp. Chapt. V, VI</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1''' , Interscience  (1963)  pp. Chapt. V, VI</TD></TR></table>

Latest revision as of 12:45, 31 July 2014

A $2$-form $\Omega$ on a principal fibre bundle $P$ with structure Lie group $G$, taking values in the Lie algebra $\mathfrak g$ of the group $G$ and defined by the connection form $\theta$ on $P$ by the formula

$$\Omega=d\theta+\frac12[\theta,\theta].$$

The curvature form is a measure of the deviation of the given connection from the locally flat connection characterized by the condition $\Omega\equiv0$. It satisfies the Bianchi identity

$$d\Omega=[\Omega,\theta]$$

and defines the holonomy algebra (see Holonomy group).


Comments

The equation $\Omega=d\theta+[\theta,\theta]/2$ is called the structure equation.

References

[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) pp. Chapt. V, VI
How to Cite This Entry:
Curvature form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curvature_form&oldid=16422
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article