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Curvature form

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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 $\theta$ 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://www.encyclopediaofmath.org/index.php?title=Curvature_form&oldid=32609
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article