View source for Curvature form

{{TEX|done}}
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 $\Omega\equiv0$. It satisfies the Bianchi identity

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

and defines the holonomy algebra (see [[Holonomy group|Holonomy group]]).



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

====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 {{ZBL|0119.37502}}</TD></TR>
</table>