= Trace of vector-bundle curvature
{title2=$\alpha=\operatorname{tr}\Theta$}
The <trace> of the <curvature form of a connection> is a global <closed differential form> of degree two. Under a frame change $e'=eg$, the <connection matrix> and curvature change by $A'=g^{-1}Ag+g^{-1}dg$ and $\Theta'=g^{-1}\Theta g$. Thus <trace> is independent of frame. The <Bianchi identity> gives $d\Theta=\Theta\wedge A-A\wedge\Theta$. Taking the <trace> cancels the two terms: one-forms commute with two-forms in the graded sense, and renaming the <matrix> indices gives equality of their <traces>. Locally $\operatorname{tr}\Theta=d\operatorname{tr}A$, since the <trace> of $A\wedge A$ vanishes by pairing its off-diagonal terms. It is the curvature of the <determinant connection>.
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