We prove the Bianchi Identity, which states that for a connection $\nabla$ on a vector bundle $\pi \colon E \to M$, its curvature satisfies

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

We then explain how the Bianchi Identity allows one to construct characteristic classes of a bundle. In this lecture we only consider a “toy” example and show how $[ \operatorname{tr}( R^{ \nabla})] \in H^2_{ \operatorname{dR}}(M)$ is a well-defined cohomology class which is independent of the connection $\nabla$.

Actually this example isn’t so interesting, since it’s always zero. ¯\_(ツ)_/¯

Next lecture we’ll construct some non-zero characteristic classes!