Connection is module structure over connection algebra

Statement

Let $E$ be a vector bundle over a differential manifold $M$. Then, a connection on $E$ is equivalent to giving $E$ the structure of a module over the connection algebra over $M$.

Definitions used

Connection

Further information: Connection

Connection algebra

Further information: Connection algebra

Proof

We start with a connection $\mathrm{\nabla }$ on $E$ and show how $\mathrm{\nabla }$ naturally equips $E$ with the structure of a module over $\mathcal{C}(M)$.

First, observe that a connection gives a rule for the Lie algebra of first-order differential operators to act on $E$, hence the tensor algebra generated by it as a vector space, acts on $E$. We need to check that under this action $m(1)-1$ acts trivially on $E$.