In this lecture we generalise differential forms by allowing them to take values in a vector space (these are called vector-valued forms), or a vector bundle (these are called bundle-valued forms).

We then explain how a connection $ \nabla$ on a vector bundle $ \pi \colon E \to M$ gives rise to a graded derivation

$$ d^{ \nabla} \colon \Omega^k(M,E)\to \Omega^{k+1}(M,E),$$

which we call an exterior covariant differential. This operator behaves similarly to the standard exterior differential $d$, save for one key difference: it does not necessarily square to zero! In fact,

$$ d^{ \nabla} \circ d^{ \nabla} \alpha = R^{\nabla} \wedge \alpha,$$

and hence $d^{ \nabla} \circ d^{ \nabla} = 0$ if and only if $ \nabla$ is flat.

We conclude by proving the Bianchi Identity, which is the following pretty formula:

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

Comments and questions?