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


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


and defines the holonomy algebra (see Holonomy group).


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


[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) pp. Chapt. V, VI Zbl 0119.37502
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This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article