Difference between revisions of "Curvature form"
From Encyclopedia of Mathematics
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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 | ||
| − | + | $$\Omega=d\theta+\frac12[\theta,\theta].$$ | |
| − | The curvature form is a measure of the deviation of the given connection from the locally flat connection characterized by the condition | + | 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|Holonomy group]]). | and defines the holonomy algebra (see [[Holonomy group|Holonomy group]]). | ||
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====Comments==== | ====Comments==== | ||
| − | The equation | + | 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"> | + | <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 {{ZBL|0119.37502}}</TD></TR> | ||
| + | </table> | ||
Latest revision as of 15:07, 19 November 2023
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 Zbl 0119.37502 |
How to Cite This Entry:
Curvature form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curvature_form&oldid=16422
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