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.


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