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 $\check{\Omega} \in \Omega^2(M,\operatorname{Ad}(P))$.