Let $ \pi \colon P \to M$ be a principal $G$-bundle with connection 1-form $ \omega \in \Omega^1(P, \mathfrak{g})$ and curvature form $Ω \in \Omega^2(P,\mathfrak{g}).$ We begin this lecture by proving Cartan’s Structure Equation for the curvature form

$$ \Omega =d \omega + \frac{1}{2}[ \omega, \omega]$$

and the Bianchi Identity

$$ d \Omega = [ \Omega ,\omega]. $$

These are the principal bundle analogues of the identities

$$ R^{\nabla} (X,Y) = \nabla_X \nabla_Y - \nabla_Y \nabla_X - \nabla_{[X,Y]}$$

and

$$ d^{\nabla^{\operatorname{End}}}R^{\nabla} = 0$$

respectively.

We also show that $ \Omega$ is horizontal and equivariant, and hence defines a bundle valued 2-form $ \check{\Omega}  \in \Omega^2(M,\operatorname{Ad}(P))$.


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