Let $\pi \colon P \to M$ be a principal $G$-bundle. We define the connection 1-form $\varpi \in \Omega^1(P, \mathfrak{g})$ associated to a connection on $P$. This is the principal bundle analogue of a covariant derivative operator $\nabla$ associated to a connection on a vector bundle.

We then define the curvature 2-form $\Omega$ associated to $\varpi$, which measures how far away the given connection is from being flat. We state Cartan’s Structure Equation, which relates $\varpi$ and $\Omega$:

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

The proof will come next lecture.