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

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

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

Now suppose $\rho \colon G \to \operatorname{GL}(V)$ is an effective smooth representation, and let $\rho(P) \to M$ denote the associated vector bundle. In Lecture 38 we showed that the connection $\varpi$ induces a connection on $\nabla$ on $\rho(P)$. Today we give an explicit formula for the exterior covariant differential $d ^{ \nabla}$ in terms of $\varpi$:

$$d^{ \nabla} \vartheta= \left( d \vartheta^{ \sharp} + \varpi \wedge_{ \rho} \vartheta^\sharp \right)^{ \flat}.$$

Isn’t that beautiful? 🤓

Finally, we show how the curvature form $\Omega^{\flat}$ is related to $R^{\nabla}$.