We then explain how the Bianchi Identity allows one to construct characteristic classes of a bundle. Along the way we introduce Riemannian metrics on vector bundles, and explain what it means for a connection to be a metric connection. We show that both metrics and metric connections always exist, and show that metric connections satisfy the so-called Ricci Identity:

$$X \langle r,s \rangle = \langle \nabla_Xr, s \rangle + \langle r, \nabla_Xs \rangle, \qquad \forall , X \in \mathfrak{X}(M), \ r,s \in \Gamma(E).$$

In this lecture we only consider a “toy” example and show how $[ \mathsf{tr}( R^{ \nabla})] \in H^2_{ \operatorname{dR}}(M)$ 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!